math: {{c1::Natural Numbers}} = {{c2::0, 1, 2, …}} or {{c2::1, 2, …}}
math: {{c1::N_0}} = {{c1::N^0}} = {{c1:: N ∪ {0} }} = {{c2:: {0, 1, 2, …} }}
math: {{c1:: N* }} = {{c1:: N+ }} = {{c1:: N1 }} = {{c1:: N>0 }} = {{c1:: N \ {0} }} = {{c2:: {1, 2, 3, …} }}
math: f is {{c1::Associative}} if {{c2::f(f(a, b), c) = f(a, f(b, c))}}
math: f is {{c1::Commutative}} if {{c2::f(a, b) = f(b, a)}}
math: In a/b, a is the {{c1::Numerator}} or {{c1::Dividend}} and b is the {{c1::Denominator}} or {{c1::Divisor}}
math: 7/3 is read as {{c1::Seven Thirds}}
math: a/b is {{c1::an Improper}} fraction if {{c2::a >= b}}
math: a/b is {{c1::a Proper}} fraction if {{c2::a < b}}
math: A {{c1::Mixed Number}} is {{c2::the sum of a non-zero integer and a proper fraction}}
math: A {{c1::Mixed Number}} is noted as {{c2::$a\frac{b}{c}$}}
math: proof: ma/mb = a/b {{c1::
Extract m/m out of the fraction:
ma/mb = a/b * m/m
= a/b * 1
= a/b
}}
- Reduce
$\frac{cx + x^2}{ca^2 + ax^2}$ to its lowest terms.\begin{align} \frac{cx + x^2}{ca^2 + ax^2} &= \frac{x(c + x)}{a^2(c + x)}
&= \frac{x}{a^2} \end{align} - Reduce
$\frac{x^3 - b^2x}{x^2 + 2bx + b^2}$ to its lowest terms.Note: I had a hard time finding this one because I didn’t immediately recognize
$x^2 - b^2 = (x + b)(x - b)$ .\begin{align} \frac{x^3 - b^2x}{x^2 + 2bx + b^2} &= \frac{x(x^2 - b^2)}{(x + b)^2}
&= \frac{x(x + b)(x - b)}{(x + b)^2} \ &= \frac{x(x - b)}{x + b} \ &= \frac{x^2 - bx}{x + b} \end{align} - Reduce
$\frac{x^4 - b^4}{x^2 + 2bx + b^2}$ to its lowest terms.Note: I had a hard time finding this one because I didn’t immediately think of factoring
$x^3$ in the denominator.\begin{align} \frac{x^4 - b^4}{x^2 + 2bx + b^2} &= \frac{(x^2 + b^2)(x^2 - b^2)}{x^3(x^2 - b^2)}
&= \frac{x^2 + b^2}{x^3} \end{align} - Reduce
$\frac{x^2 - y^2}{x^4 - y^4}$ to its lowest terms.\begin{align} \frac{x^2 - y^2}{x^4 - y^4} &= \frac{(x + y)(x - y)}{(x^2 + y^2)(x^2 - y^2)}
&= \frac{(x + y)(x - y)}{(x^2 + y^2)(x + y)(x - y))} \ &= \frac{1}{x^2 + y^2} \end{align} - Reduce
$\frac{a^4 - x^4}{a^3 - a^2x - ax^2 + x^3}$ to its lowest terms.\begin{align} \frac{a^4 - x^4}{a^3 - a^2x - ax^2 + x^3} &= \frac{(a^2 + x^2)(a^2 - x^2)}{(a^2 - x^2)(a - x)}
&= \frac{a^2 + x^2}{a - x} \end{align} - Reduce
$\frac{5a^5 + 10a^4x + 5a^3x^2}{a^3x + 2a^2x^2 + 2ax^3 + x^4}$ to its lowest terms.\begin{align} \frac{5a^5 + 10a^4x + 5a^3x^2}{a^3x + 2a^2x^2 + 2ax^3 + x^4} &= \frac{5a^3(a^2 + 2ax + x^2)}{(a + x)(a^2x + ax^2 + x^2)}
&= \frac{5a^3(a + x)^2}{(a + x)(a^2x + ax^2 + x^2)} \ &= \frac{5a^3(a + x)}{a^2x + ax^2 + x^2} \ &= \frac{5a^4 + 5a^3x}{a^2x + ax^2 + x^2} \end{align}
math: a/c + b/c = {{c1::(a + b)/c}} because {{c1::
a/c + b/c = a * 1/c + b * 1/c
= (a + b) * 1/c
= (a + b)/c
}}
- Reduce
$\frac{2x}{a}$ and$\frac{b}{c}$ to a common denominator.\begin{align} \frac{2x}{a} &= \frac{2cx}{ac}
\frac{b}{c} &= \frac{ab}{ac} \end{align} - Reduce
$\frac{a}{b}$ and$\frac{a + b}{c}$ to a common denominator.\begin{align} \frac{a}{b} &= \frac{ac}{bc}
\frac{a + b}{c} &= \frac{ab + b^2}{bc} \end{align} - Reduce
$\frac{3x}{2a}$ ,$\frac{2b}{3c}$ and$d$ to a common denominator.\begin{align} \frac{3x}{2a} &= \frac{9cx}{6ac}
\frac{2b}{3c} &= \frac{4ab}{6ac} \ d &= \frac{6acd}{6ac} \end{align} - Reduce
$\frac{3}{4}$ ,$\frac{2x}{3}$ and$a + \frac{2x}{a}$ to a common denominator.\begin{align} \frac{3}{4} &= \frac{9a}{12a}
\frac{2x}{3a} &= \frac{8ax}{12a} \ a + \frac{2x}{a} &= \frac{12a^2 + 24x}{12a} \end{align} - Reduce
$\frac{1}{2}$ ,$\frac{a^2}{3}$ and$\frac{x^2 + a^2}{x + a}$ to a common denominator.\begin{align} \frac{1}{2} &= \frac{3(x + a)}{6(x + a)}
\frac{a^2}{3} &= \frac{2a^2(x + a)}{6(x + a)} \ \frac{6(x^2 + a^2)}{x + a} &= \frac{6(x^2 + a^2)}{6(x + a)} \end{align} - Reduce
$\frac{b}{2a^2}$ ,$\frac{c}{2a}$ and$\frac{d}{a}$ to a common denominator.\begin{align} \frac{b}{2a^2} &= \frac{b}{2a^2}
\frac{c}{2a} &= \frac{ac}{2a^2} \ \frac{d}{a} &= \frac{2ad}{2a^2} \end{align}
math: (a/b)/c = {{c1::a/bc}}
math: proof: (a/b)/c = a/bc {{c1::
(a/b)/c = a/b * 1/c
= a/bc
}}
math: proof: 1/ab = 1/a * 1/b {{c1::
1/ab is the inverse of ab, so
ab * 1/ab must be equal to 1
ab * (1/a * 1/b) = a * 1/a * b * 1/b
= 1 * 1
= 1
}}
math: proof: (a/b) * (c/d) = ac/bd {{c1::
(a/b) * (c/d) = a * 1/b * c * 1/d
= a * c * 1/b * 1/d
= a * c * 1/bd (multiplicative inverse is the product of multiplicative inverses)
= ac/bd
}}
math: {{c1:: least_common_multiple(a,b) }} = {{c2::the smallest number that is divisible by both a and b}}
math: least_common_multiple(4, 6) = {{c1::12}}
math: {{c1:: greatest_common_divisor(a,b) }} = {{c2::the greatest number that divides both a and b}}
math: greatest_common_divisor(8, 12) = {{c1::4}}
math: To reduce a fraction {{c1::divide the numerator and denominator by their greatest common divisor}}
math: a/b is an Irreducible Fraction if {{c1:: greatest_common_divisor(a, b) = 1 }}
math: {{c1::Irreducible Fraction}} aka {{c2::Simplified Fraction}}
math: {{c1::Multiplicative Inverse}} aka {{c2::Reciprocal}}
math: Y is the {{c1::Muliplicative Inverse}} of X if {{c2::X * Y = 1}}
math: The multiplicative inverse of {{c1::a/b}} is {{c2::b/a}}
math: proof: The multiplicative inverse of a/b is b/a: {{c1::
State as an equation, and then cancel the denominator:
a/b * X = 1
X = 1/(a/b)
X = 1/(a/b) * 1
X = 1/(a/b) * (b/a)/(b/a)
X = (b/a)/(ab/ab)
X = (b/a)/1 (identity property of division)
X = b/a
}}
math: 1/(a/b) = {{c1::b/a}}
math: proof: 1/(a/b) = b/a: {{c1::
Cancel the demoninator by multipying the numerator and the denominator by b/a:
1/(a/b) = 1/(a/b) * 1
= 1/(a/b) * (b/a)/(b/a)
= (b/a)/(ab/ab)
= (b/a)/1
= b/a
}}
math: (a/b)/(c/d) = {{c1::ad/bc}}
math: proof: (a/b)/(c/d) = ad/bc {{c1::
(a/b)/(c/d) = (a/b) * 1/(c/d) (division is multiplication by inverse)
= (a/b) * (d/c) (inverse)
= ad/bc
}}
math: A Repeating Decimal is a {{c1::Decimal Number in which the last digits repeat endlessly}}
math: A Repeating Decimal is written as {{c1::$X.Y\overline{Z}$}}
math: {{c1::Mean}} aka {{c2::Arithmetic Average}}
math: {{c1::Mean}} = {{c2::Sum of all the Values / The Number of Values}}
math: {{c1::Median}} = {{c2::The middle value when there is an odd number of them. The mean of the two middle values otherwise}}
math: {{c1::Mode}} = {{c2::The number with the highest frequency}}
math: Probability Formula = {{c1::Number of Favorable Outcomes/Number of Outcomes}}
math: A {{c1::Ratio}} is {{c2::a fraction of two numbers with the same unit}}
math: A {{c1::Rate}} is {{c2::a fraction of two numbers with different units}}
math: A {{c1::Unit Rate}} is {{c2::a Rate with a denominator of 1}}
math: A {{c1::Perfect Square}} is a {{c2::Square of an integer}}
math: {{c1::Radical Sign}} aka {{c2::Root Symbol}}
math:
math: {{c1::Rational Numbers}} are {{c2::numbers that can be expressed as a fraction of two integers}}
math: {{c1::Irrational Numbers}} are {{c2::all the numbers that aren’t rational}}
math: {{c1::Real Numbers}} are numbers that are {{c2::either rational or irrational}}
math: {{c1::Natural Numbers Symbol}} = {{c2::ℕ}}
math: {{c1::Integers Symbol}} = {{c2::ℤ}}
math: {{c1::Rational Numbers Symbol}} = {{c2::ℚ}}
math: {{c1::Real Numbers Symbol}} = {{c2::ℝ}}
math: {{c1::Real}} Numbers include {{c1::Rational Numbers}} wich include {{c1::Integers}} which include {{c1::Natural Numbers}}
f is left-distributive over g if f(a, g(b, c)) = g(f(a, b), f(a, c))
f is right-distributive over g if f(g(b, c), a) = g(f(b, a), f(c, a))
math: x is the {{c1::identity element}} of f if {{c2::f(x, y) = y}}
math: Two angles are {{c1::Supplementary}} if {{c2::their sum is 180°}}
math: Two angles are {{c1::Complementary}} if {{c2::their sum is 90°}}
math: Two figures are {{c1::Similar}} if {{c2::they have the same shape}}
math: Two Similar Triangles {{c1::have the same}} angles
math: Two figures are {{c1::Congruent}} if {{c2::they have the same shape and size}}
math: Triangle Area = {{c1::1/2 * (base * height)}}
math: The sum of a triangle angles = {{c1::180°}}
math: A {{c1::Trapezoid}} is {{c2::a quadrilateral with at least one pair of parallel sides}}
math: Trapezoid Area = {{c1::1/2 * height * (big base + small base)}}
math: The Area of a Trapezoid can be thought of as {{c1::the area of two triangles}}
math: Circle Circumference = {{c1::2πr}}
math: Circle Area = {{c1::πr^2}}
math: Sphere Surface Area = {{c1::4πr^2}}
math: Sphere Volume = {{c1::4/3 * πr^3}}
math:
- Circle Circumference = {{c1::$2 π r$}}
- Circle Area = {{c1::$π r^2$}}
- Sphere Surface = {{c1::$4 π r^2$}}
- Sphere Volume = {{c1::$\frac{4}{3} π r^3$}}
math: Cylinder Surface Area = {{c1::2 * surface of base + surface of rectangle = 2πr^2 + 2πr * height}}
math: Cylinder Volume = {{c1::surface of base * height = πr^2 * height}}
math: Cone Volume = {{c1::1/3 * surface of base * height = 1/3 * πr^2 * height}}
math: The degree of a Polynomial is the {{c1::highest degree of all its terms}}
math: The Standard Form of a Polynomial lists the terms {{c1::in decreasing order of degrees}}
math: {{c1::a^m * a^n}} = {{c2::a^(m + n)}}
math: {{c1::(a^m)^n}} = {{c2::a^(mn)}}
math: {{c1::(ab)^m}} = {{c2::a^m * b^m}}
math: {{c1::a^m / a^n}} = {{c2::a^(m - n)}}
math: {{c1::(a / b)^m}} = {{c2::a^m / b^m}}
math: {{c1::a^-n}} = {{c2::1 / a^n}}
math: {{c1::An Intercept of a Line}} is {{c2::a point where the line crosses the x or y-axis}}
math: An x-intercept is a point where a line crosses the {{c1::x-axis}}
math: A y-intercept is a point where a line crosses the {{c1::y-axis}}
math: Formula of the Slope of a Line = {{c1::Rise/Run}} or {{c1::(y2 - y1)/(x2 - x1)}} or {{c1::m in y = mx + b}}
math: Using the Prime Factorization, the Least Common Multiple of two numbers is {{c1::the multiplication of their prime factors to their highest powers}} Example lcm(8, 18, 21) = {{c1::2^3 * 3^2 * 7^1}}
math: {{c1::An equation that is true for some values of the variable (eg. 2 x = 4)}} is a {{c2::Conditional Equation}}
math: {{c1::An equation that is true for any value of the variable (eg. x + 3 = x + 3)}} is {{c2::an Identity Equation}}
math: {{c1::An equation that is false for all values of the variable (eg. x = x + 1)}} is {{c2::a Contradiction Equation}}
math: To easily simplify equations with fractions {{c1::multiply both sides of the equation by the LCD of all the fractions (you need to multiply by a number that is divisible by all the denominators)}}
math: An {{c1::Interval Notation}} is written as {{c2::[X, Y] when X and Y are included or (X, Y) otherwise}}
math: When you divide or multiply an inequality by a negative number, the inequality {{c1::reverses}}
math: When you divide or multiply an inequality by a positive number, the inequality {{c1::stays the same}}
math: Solve -10x >= 50 and give the solution as an interval notation {{c1::
-10x >= 50 x =< -5 Solution: (-∞, -5]
}}
math: Translate the following to an Inequality: Bob plans to rent a car from a company that charges $75 a week plus $0.25 a mile. How many miles can he travel and still keep within his $200 budget? {{c1:: 75 + 0.25m <= 200 }}
math: {{c1::Parallel}} lines have {{c2::the same}} slope
math: {{c1::Perpendicular}} lines have {{c2::slopes that are the negative reciprocals of each other::slope}}
math: inequality graph: A {{c1::Dashed}} Boundary Line means the line {{c2::isn’t}} included in the solution
math: inequality graph: A {{c1::Solid}} Boundary Line means the line {{c2::is}} included in the solution
math: {{c1::Coincident Lines}} have {{c2::the same slope and y-intercept}}
math: {{c1::A Consistent}} System of Equations has {{c2::at least one}} solution
math: {{c1::An Inconsistent}} System of Equations has {{c2::no solution}}
math: {{c1::Dependent}} Equations have {{c2::the same}} solutions
math: {{c1::Independent}} Equations have {{c2::different}} solutions
math: Solve the following system by Substitution:
x + 2y = 4
x + 3y = 2
{{c1::
x = 4 - 2y (isolate x in the first equation)
4 - 2y + 3y = 2 (substitute x in the second equation)
y = -2
x + 2(-2) = 4 (use y in one of the equations (here the first))
x = 8
8 + 2(-2) = 4 (check that the solution is correct in the first equation)
4 = 4
8 + 3(-2) = 2 (check that the solution is correct in the second equation)
2 = 2
Solution: (8, -2)
}}
math: Solving Systems of Equations by Elimination is based on the {{c1::addition/substraction property of equality}}
math: Solve the following system by Elimination:
x + 3y = 2
x + 2y = 4
{{c1::
x + 3y - x - 2y = 2 - 4 (substract the second equation from the first one)
y = -2
x + 2(-2) = 4 (use y in one of the equations)
x = 8
8 + 3(-2) = 2 (check that the solution is correct in the first equation)
2 = 2
8 + 2(-2) = 4 (check that the solution is correct in the second equation)
4 = 4
Solution: (8, -2)
}}
math: The degree of a Polynomial Term is {{c1::the sum of the exponents of its variables}}
math: pattern: {{c1::(a + b)^2}} = {{c2::a^2 + 2ab + b^2}}
math: pattern: {{c1::(a - b)^2}} = {{c2::a^2 - 2ab + b^2}}
math: A {{c1::Conjugate Pair}} is {{c2::two binomials of the form (a + b),(a - b)::form}}
math: pattern: {{c1::(a + b)(a - b)}} = {{c2::a^2 - b^2}}
math: Divide (2x^2 + 4) by (x - 2): {{c1::
2x + 4 + 12/(x - 2)
____________________
x - 2 | 2x^2 + 0x + 4
-(2x^2 - 4x)
4x + 4
-(4x - 8)
12
}}
math: {{c1::a^-n}} = {{c2::1/a^n}}
math: 1/a^-n = {{c1::a^n}}
math: {{c1::(a/b)^-n}} = {{c2::(b/a)^n}}
math: Factor 4y^2 + 24y + 28 {{c1::
Factor out the GCF: 4y^2 + 24y + 28 = 4(y^2 + 6y + 7)
}}
math: Factor xy + 3y + 2x + 6 {{c1::
Factor by grouping:
xy + 3y + 2x + 6 = y(x + 3) + 2(x + 3)
= (y + 2)(x + 3)
}}
math: Factor x^2 - 11x + 24 {{c1::
x^2 + bx + c = (x + m)(x + n) with b = m + n and c = m * n b = -11 = -3 + -8 c = 24 = -3 * -8 So x^2 - 11x + 24 = (x - 3)(x - 8)
}}
math: Factor x^2 + 4x - 5 {{c1::
x^2 + bx + c = (x + m)(x + n) with b = m + n and c = m * n b = 4 = 5 + -1 c = -5 = 5 * -1 So x^2 + 4x - 5 = (x + 5)(x - 1)
}}
math: {{c1::A Prime Polynomial}} is {{c2::a Polynomial that cannot be factorized}}
math: {{c1::Prime Polynomial}} aka {{c2::Irreducible Polynomial}}
math: Factor y^2 − 6y + 15 {{c1::
y^2 − 6y + 15 is irreducible over the integers
}}
math: Factor x^2 - 8xy + 15y^2 {{c1::
x^2 - 8xy + 15y^2 = (x + m)(x + n) with b = m + n and c = m * n b = 8y = -5y - 3y c = 15y^2 = -5y * -3y So x^2 - 8xy + 15y^2 = (x - 5y)(x - 3y)
}}
math: Factor 6x^2 + 7x + 2 {{c1::
Factor with the ac method: 6x^2 + 7x + 2 = ax^2 + mx + nx + c with b = m + n and a * c = m * n ac = 12 b = 7 Possible factors of 12 are: 12 * 1 > 12 + 1 = 13 6 * 2 > 6 + 2 = 8 4 * 3 > 4 + 3 = 7 = b So 6x^2 + 7x + 2 = 6x^2 + 4x + 3x + 2 Factor by grouping: 6x^2 + 4x + 3x + 2 = 3x(2x + 1) + 2(2x + 1) = (3x + 2)(2x + 1)
}}
math: pattern: {{c1::a^3 + b^3}} = {{c2::(a + b)(a^2 - ab + b^2)}}
math: pattern: {{c1::a^3 - b^3}} = {{c2::(a - b)(a^2 + ab + b^2)}}
math: Factor 4n^2 + 12n + 9 {{c1::
4n^2 + 12n + 9 is of the form a^2 + 2ab + b^2 = (a + b)^2 4n^2 + 12n + 9 = (2n + 3)^2
}}
math: Factor 100x^2 − 20x + 1 {{c1::
100x^2 − 20x + 1 is of the form a^2 - 2ab + b^2 = (a - b)^2 100x^2 − 20x + 1 = (10x - 1)^2
}}
math: Factor 27q^2 - 3 {{c1::
27q^2 - 3 = 3(9q^2 - 1) 9q^2 - 1 is of the form a^2 - b^2 = (a + b)(a - b) 3(9q^2 - 1) = 3(3q + 1)(3q - 1)
}}
math: Factor x^3 + 64 {{c1::
x^3 + 64 is of the form a^3 + b^3 = (a + b)(a^2 - ab + b^2) x^3 + 64 = (x + 4)(x^2 - 4x + 16)
}}
math: Factor 27 - 125p^3 {{c1::
27 - 125p^3 is of the form a^3 - b^3 = (a - b)(a^2 + ab + b^2) 27 - 125p^3 = (3 - 5p)(9 + 15p + 25p^2)
}}
math: Factor m^4 - n^4 {{c1::
m^4 - n^4 is of the form a^2 - b^2 = (a + b)(a - b) m^4 - n^4 = (m^2 + n^2)(m^2 - n^2) m^2 - n^2 is of the form a^2 - b^2 = (a + b)(a - b) (m^2 + n^2)(m^2 - n^2) = (m^2 + n^2)(m + n)(m - n)
}}
math: Factor 9x^2 + 4 {{c1::
9x^2 + 4 is irreducible over the integers
}}
math: General Strategy for Factoring Polynomials {{c1::
- Factor out the GCF if there is one
- For binomials:
- If of the form a^2 + b^2: Do nothing
- If of the form a^2 - b^2: (a + b)(a - b)
- If of the form a^3 + b^3: (a + b)(a^2 - ab + b^2)
- If of the form a^3 - b^3: (a - b)(a^2 + ab + b^2)
- For trinomials:
- If of the form x^2 + bx + c: (x + m)(x + n) where b = m + n and c = m * n
- If of the form ax^2 + bx + c:
- If a and c are perfect squares:
- If of the form a^2 + 2ab + b^2: (a + b)^2
- If of the form a^2 - 2ab + b^2: (a - b)^2
- The ac method: ax^2 + mx + nx + c with b = m + n and a * c = m * n, then use grouping
- Or use the trial and error method
- If it has more than 3 terms: Use grouping
}}
math: A {{c1::Quadratic Equation}} is an equation of the form {{c2::ax^2 + bx + c = 0}}
math: Solve a Quadratic Equation by Factoring by:
- {{c1::Writing it in standard form}}
- {{c1::Factoring it completely}}
- {{c1::Using the Zero Product property}}
- {{c1::Solving the linear equations}}
math: Solve 3c^2 = 10c − 8 by factoring {{c1::
Write in standard form: 3c^2 - 10c + 8 = 0 Factor (with the ac method): ac = 24 = 6 * 4 b = 10 = 6 + 4 3c^2 - 6c - 4c + 8 = 0 3c(c - 2) - 4(c - 2) = 0 (3c - 4)(c - 2) = 0 Use the Zero Product property: If (3c - 4)(c - 2) = 0 then either 3c - 4 = 0 or c - 2 = 0 or both. Solve the linear equations: 3c - 4 = 0 3c = 4 c = 4/3 c - 2 = 0 c = 2 Solutions: c = 4/3 or c = 2
}}
math: A {{c1::Rational Expression}} is {{c2::an expression of the form f(x)/g(x) where f and g are polynomials}}
math: {{c1::Rational Expression}} aka {{c2::Rational Fraction}}
math: (a - b)/(b - a) = {{c1::(a - b)/-(a - b)}} = {{c1::-1}}
math: Find the Least Common Multiple of polynomials by {{c1::factoring them completely and multiplying their factors to their highest powers}} Example: lcm(a^2 - b^2, (a + b)^2) = {{c1::(a - b)(a + b)^2}}
math: To add rational expressions with unlike denominators:
- {{c1::Find the Least Common Denominator}}
- {{c1::Multiply the fractions’ denominator and numerator by the factors their denominator was missing}}
- {{c1::Add the rational expressions now that they have the same denominators}}
math: Simplify
cd + 3c = c(d + 3) \
d^2 - 9 = (d + 3)(d - 3) \
\
\text{Find the least common denominator} \
lcm(cd + 3c, d^2 - 9) = c(d + 3)(d - 3) \
\
\text{Multiply the numerators and denominators by} \
\text{the factors the denominators were missing} \
\frac{4(d - 3)}{c(d + 3)(d - 3)} + \frac{c}{c(d + 3)(d - 3)} \
\
\text{Add} \
\frac{4(d - 3) + c}{c(d + 3)(d - 3)} \
\end{gather*}
}}
math: A {{c1::Complex Rational Expression}} is a {{c2::rational expression whose numerator or denominator contains a rational expression}}
math: Simplify a Complex Rational Expression by either:
- {{c1::Multiplying the numerator by the reciprocal of the denominator}}
- {{c1::Multiplying the numerator and denominator by the LCD of all the fractions}}
Example:
{{c1::
\begin{gather*}
\text{Multiplying the numerator by the reciprocal of the denominator}
\frac{ \frac{a}{b} }
{ \frac{x}{y} } \
\frac{a}{b} ⋅ \frac{y}{x} \
\frac{ay}{bx} \
\text{Multiplying the numerator and denominator by the LCD of all the fractions} \
\frac{ \frac{a}{b} }
{ \frac{x}{y} } \
\frac{\frac{a}{b} ⋅ by}
{\frac{x}{y} ⋅ by} \
\frac{ay}{bx}
\end{gather*}
}}
math: Simplify
\frac{ \frac{y + x}{xy} }
{ \frac{x^2 - y^2}{xy} } \
\text{Multiply the numerator by the reciprocal of the denominator} \
\frac{(y + x)xy}{xy(x^2 - y^2)} \
\text{Simplify} \
\frac{1}{x - y}
\end{gather*}
}}
math: Simplify
\frac{ \color{red}{xy} ⋅ \frac{1}{x} + \color{red}{xy} ⋅ \frac{1}{y} }
{ \color{red}{xy} ⋅ \frac{x}{y} - \color{red}{xy} ⋅ \frac{y}{x} } \
\text{Simplify} \
\frac{y + x}{x^2 - y^2} \
\frac{1}{x - y}
\end{gather*}
}}
math: {{c1::An Extraneous Solution}} to an equation is {{c2::a solution that would make the equation be undefined}}
math: A {{c1::Proportion}} is an equation of the form {{c2::a/b = c/d}}
math: A proportion a/b = c/d is read as {{c1::a is to b as c is to d}}
math: Bob can paint a room in 6 hours. Alice takes 12 hours to paint the same room. How long would it take Bob and Alice to paint the room if they worked together? {{c1::
Translate to an equation: 1/6 + 1/12 = 1/x Multiply by the LCD of all the fractions to remove them: 12x(1/6 + 1/12) = 12x*1/x 2x + x = 12 Simplify: 3x = 12 x = 12/3 x = 4 Answer: 4 hours
}}
math: x varies directly with y if {{c1::x = ky}}
math: The distance a moving body travels, d, varies directly with the time, t, it moves. A train travels 100 miles in 2 hours. Write the equation that relates d and t. {{c1::
d = kt 100 = k2 k = 50 Answer: d = 50t
}}
How many miles would it travel in 5 hours? {{c1::
miles = kt = 50 * 5 = 250 miles
}}
math: x varies inversely with y if {{c1::x = k/y}}
math: The time required to empty a tank varies inversely with the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute). Write the equation that relates the number of hours to the pump rate. {{c1::
time = k/rate 5 = k/200 k = 5 * rate = 1000 Answer: time = 1000/rate
}}
How long would it take Janet to pump her basement if she used a pump rated at 400 gpm? {{c1::
time = 1000/rate = 1000/400 = 2.5
}}
math: {{c1::$\sqrt{ab}$}} = {{c2::$\sqrt{a}\sqrt{b}$}}
math: {{c1::$\sqrt{ \frac{a}{b} }$}} = {{c2::$\frac{ \sqrt{a} }{ \sqrt{b} }$}}
math: Simplify and rationalize
\frac{ \sqrt{11} }{ 2\sqrt{7} } \
\frac{ \sqrt{11} ⋅ \color{red}{ \sqrt{7} } }{ 2\sqrt{7} ⋅ \color{red}{ \sqrt{7} } } \
\frac{ \sqrt{77} }{2 ⋅ 7} \
\frac{ \sqrt{77} }{14}
\end{gather*}
}}
math: Simplify
\frac{ 4\color{red}{(4 - \sqrt{2})} }{ (4 + \sqrt{2}) \color{red}{(4 - \sqrt{2})} } \
\frac{4(4 - \sqrt{2})}{16 - 2} \
\frac{4(4 - \sqrt{2})}{14} \
\frac{2(4 - \sqrt{2})}{7}
\end{gather*}
}}
math: Solve
5n - 4 = 81 \
5n = 85 \
n = 17
\end{gather*}
}}
math: Solve
p - 1 = (p - 1)^2 \
p - 1 = p^2 - 2p + 1 \
0 = p^2 - 3p + 2 \
0 = (p - 2)(p - 1) \
p = 2 \text{ or } p = 1 \
\
\text{Check for extraneous solutions:} \
\sqrt{2 - 1} + 1 = 2 \
1 + 1 = 2 \text{, OK} \
\
\sqrt{1 - 1} + 1 = 1 \
0 + 1 = 1 \text{, OK} \
\
\text{Solution: p = 2 or p = 1}
\end{gather*}
}}
math: Solve
\text{Since the square root is a negative number, there is no real solution.}
\end{gather*}
}}
math: Solve
r + 4 = r^2 - 4r + 4 \
0 = r^2 - 5r \
0 = r(r - 5) \
r = 0 \text{ or } r = 5 \
\
\text{Check for extraneous solutions:} \
\sqrt{0 + 4} - 0 + 2 = 0 \
2 + 2 = 0 \
4 ≠ 0 \text{, 0 is an extraneous solution} \
\
\sqrt{5 + 4} - 5 + 2 = 0 \
3 - 5 + 2 = 0 \
0 = 0 \text{, OK} \
\
\text{Solution: r = 5}
\end{gather*}
}}
math: If {{c1::a^n = b}} then a is {{c2::an nth root of b}}
math: In
math: State the type of the root of
- If n is even - and x >= 0 then the root is a real number - and x < 0 then the root is an imaginary number - If n is odd - The root is always a real number
}}
math: $\sqrt[4]{x12}$ = {{c1::$|x^3|$}}
math:
math:
math: {{c1::$\sqrt[n]{a}$}} (root form) = {{c2::$a^\frac{1}{n}$}} (exponent form)
math: Simplify $16-\frac{3{2}}$
{{c1::
\begin{gather*}
\frac{1}{ 16^\frac{3}{2} }
\frac{1}{64}
\end{gather*}
}}
math: Solve
x &= ± \sqrt{48} \
x &= ± 4 \sqrt{3}
\end{align*}
}}
math: Solve
x = ± \sqrt{-24} \
\text{There is no real solution}
\end{gather*}
}}
math: Solve
x - 2 = ± \sqrt{27} \
x - 2 = ± 3 \sqrt{3} \
x = 2 ± 3 \sqrt{3}
\end{gather*}
}}
math: Solve
(p - 5)^2 &= 18 \
p - 5 &= ± 3 \sqrt{2} \
p &= 5 ± 3 \sqrt{2}
\end{align*}
}}
math: Solve
y^2 - 6y = 16 \
y^2 - 2 ⋅ 3y + 9 = 16 + 9 \
(y - 3)^2 = 25 \
y - 3 = ± \sqrt{25} \
y - 3 = ± 5 \
y = 3 ± 5 \
y = -2, y = 8
\end{gather*}
}}
math: Solve
2(x^2 - \frac{3}{2}x) = 20 \
x^2 - \frac{3}{2}x = 10 \
x^2 - 2\frac{3}{4}x = 10 \
x^2 - 2\frac{3}{4}x + \frac{9}{16} = 10 + \frac{9}{16} \
x^2 - 2\frac{3}{4}x + \frac{9}{16} = \frac{169}{16} \
(x - \frac{3}{4})^2 = \frac{169}{16} \
x - \frac{3}{4} = ± \frac{13}{4} \
x = \frac{3}{4} ± \frac{13}{4} \
x = 4, x = -\frac{5}{2}
\end{gather*}
}}
math: The solution to
math: Derive the Quadratic Formula
{{c1::
\begin{gather*}
\text{Complete the square}
ax^2 + bx + c = 0 \
ax^2 + bx = -c \
x^2 + \frac{b}{a}x = \frac{-c}{a} \
x^2 + \frac{b}{a}x + (\frac{1}{2}\frac{b}{a})^2 = \frac{-c}{a} + (\frac{1}{2}\frac{b}{a})^2 \
x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = \frac{-c}{a} + \frac{b^2}{4a^2} \
(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2} \
\
\text{Solve for x} \
x + \frac{b}{2a} = ± \sqrt{ \frac{b^2 - 4ac}{4a^2} } \
x + \frac{b}{2a} = ± \frac{ \sqrt{b^2 - 4ac} }{2a} \
x = - \frac{b}{2a} ± \frac{ \sqrt{b^2 - 4ac} }{2a} \
x = \frac{ -b ± \sqrt{b^2 - 4ac} }{2a}
\end{gather*}
}}
math: Solve
3u^2 + 4u - 2 = 0 \
\
\text{Use the quadratic formula} \
u = \frac{ -b ± \sqrt{b^2 - 4ac} }{2a} \
u = \frac{ -4 ± \sqrt{16 + 24} }{6} \
\
\text{Simplify} \
u = \frac{ -4 ± \sqrt{40} }{6} \
u = \frac{ -4 ± 2 \sqrt{10} }{6} \
u = \frac{ -2 ± \sqrt{10} }{3}
\end{gather*}
}}
math: In the Quadratic Formula, {{c1::$b^2 - 4ac$}} is {{c2::the Discrimant}}
math: State the number of solutions of a Quadratic Equation based on its Discriminant values:
- If {{c1::$b^2 - 4ac > 0$, the equation has two solutions}}
- If {{c1::$b^2 - 4ac = 0$, the equation has one solution}}
- If {{c1::$b^2 - 4ac < 0$, the equation has no real solutions}}
The discrimant is the value in the square root in the Quadratic Formula. If it’s positive the Formula has two real solutions. If it’s equal to zero the Formula has only one solution. If it’s negative the Formula has no real solutions.
math: Solve a Quadratic Equation by:
- {{c1::Trying to factor it first}}
- {{c1::Next trying to use the square root property}}
- {{c1::Finally using the quadratic formula}}
math: For a Quadratic Equation
- {{c1::Upward if
$a > 0$ }} - {{c1::Downward if
$a < 0$ }}
math: The {{c1::Vertex}} of a Parabola is {{c2::its highest or lowest}} point
math: Parabola’s {{c1::Vertex/Axis of Symmetry}} x-coordinate = {{c2::$-\frac{b}{2a}$}}
math: For the parabola
xvertex = -\frac{b}{2a} \
xvertex = -\frac{-6}{6} = 1 \
\
\text{Now replace
math: To find the x-intercepts of a Parabola, {{c1::let y = 0 and solve for x}} To find the y-intercept of a Parabola, {{c1::let x = 0 and solve for y}}
math: Find the intercepts of the parabola
y = 0^2 - 2*0 - 8 = -8 \
\text{y-intercept: (0, -8)} \
\
\text{For the x-intercept, let y = 0 and solve for x} \
0 = x^2 - 2x - 8 \
0 = (x - 4)(x + 2) \
x = 4, x = -2 \
\text{x-intercepts: (4, 0), (-2, 0)}
\end{gather*}
}}
math: State the number of x-intercepts of a Parabola based on its Discriminant values:
- If {{c1::$b^2 - 4ac > 0$, the parabola has two x-intercepts}}
- If {{c1::$b^2 - 4ac = 0$, the parabola has one x-intercept}}
- If {{c1::$b^2 - 4ac < 0$, the parabola has no x-intercept}}
math: The {{c1::y-coordinate of the vertex}} of the graph of a quadratic equation is the:
- {{c2::minimum value of the equation if the parabola opens upward}}
- {{c2::maximum value of the equation if the parabola opens downward}}
math: Find the minimum or maximum value of the quadratic equation
\text{A minimum value is to be found.} \
\
\text{The minimum value is at the Vertex} \
xvertex = -\frac{b}{2a} = -\frac{2}{2} = -1 \
yvertex = 1 - 2 - 8 = -9 \
\
\text{The minimum value is