Basic Properties & Facts
Arithmetic Operations
a b plus a c equals a times the quantity b plus c
\[ ab + ac = a(b+c) \]
a times the fraction b over c, equals a c over b
\[ a\!\left(\dfrac{b}{c}\right) = \dfrac{ac}{b} \]
the fraction a over b, divided by c, equals a over b c
\[ \dfrac{\,\dfrac{a}{b}\,}{c} = \dfrac{a}{bc} \]
a over b plus c over d equals the quantity a d plus b c, over b d
\[ \dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad+bc}{bd} \]
a over b minus c over d equals the quantity a d minus b c, over b d
\[ \dfrac{a}{b} - \dfrac{c}{d} = \dfrac{ad-bc}{bd} \]
a minus b over c minus d equals b minus a over d minus c
\[ \dfrac{a-b}{c-d} = \dfrac{b-a}{d-c} \]
a b plus a c all over a equals b plus c, provided a is not equal to 0
\[ \dfrac{ab+ac}{a} = b+c, \quad a \neq 0 \]
the quantity a plus b over c equals a over c plus b over c
\[ \dfrac{a+b}{c} = \dfrac{a}{c} + \dfrac{b}{c} \]
the fraction a over b, divided by the fraction c over d, equals a d over b c
\[ \dfrac{\,\dfrac{a}{b}\,}{\,\dfrac{c}{d}\,} = \dfrac{ad}{bc} \]
Exponent Properties
a to the power of n, times a to the power of m, equals a to the power of n plus m
\[ a^n a^m = a^{n+m} \]
a to the power of n, over a to the power of m, equals a to the power of n minus m, which equals 1 over a to the power of m minus n
\[ \dfrac{a^n}{a^m} = a^{n-m} = \dfrac{1}{a^{m-n}} \]
the quantity a to the power of n, raised to the power of m, equals a to the power of n times m
\[ \left(a^n\right)^m = a^{nm} \]
a to the power of 0 equals 1, provided a is not equal to 0
\[ a^0 = 1, \quad a \neq 0 \]
the quantity a b, to the power of n, equals a to the power of n times b to the power of n
\[ (ab)^n = a^n b^n \]
the quantity a over b, to the power of n, equals a to the power of n over b to the power of n
\[ \left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n} \]
a to the negative n equals 1 over a to the power of n
\[ a^{-n} = \dfrac{1}{a^n} \]
the quantity a over b, to the negative n, equals the quantity b over a, to the power of n, which equals b to the power of n over a to the power of n
\[ \left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^n = \dfrac{b^n}{a^n} \]
a to the power of one over n equals the nth root of a, which also equals the quantity a to the power of n, to the power of one over n
\[ a^{1/n} = \sqrt[n]{a} = \left(a^n\right)^{1/n} \]
Properties of Radicals
the nth root of a equals a to the power of one over n
\[ \sqrt[n]{a} = a^{1/n} \]
the nth root of a b equals the nth root of a times the nth root of b
\[ \sqrt[n]{ab} = \sqrt[n]{a}\,\sqrt[n]{b} \]
the m-th root of the nth root of a equals the m times n-th root of a
\[ \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a} \]
the nth root of a over b equals the nth root of a, over the nth root of b
\[ \sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} \]
the nth root of a to the power of n equals a, if n is odd
\[ \sqrt[n]{a^n} = a \quad \text{if } n \text{ is odd} \]
the nth root of a to the power of n equals the absolute value of a, if n is even
\[ \sqrt[n]{a^n} = |a| \quad \text{if } n \text{ is even} \]
Properties of Inequalities
If a is less than b, then a plus c is less than b plus c, and a minus c is less than b minus c
\[ \text{If } a < b \text{ then } a+c < b+c \text{ and } a-c < b-c \]
If a is less than b and c is greater than 0, then a times c is less than b times c, and a over c is less than b over c
\[ \text{If } a < b \text{ and } c > 0 \text{ then } ac < bc \text{ and } \dfrac{a}{c} < \dfrac{b}{c} \]
If a is less than b and c is less than 0, then a times c is greater than b times c, and a over c is greater than b over c
\[ \text{If } a < b \text{ and } c < 0 \text{ then } ac > bc \text{ and } \dfrac{a}{c} > \dfrac{b}{c} \]
Properties of Absolute Value
the absolute value of a equals a if a is greater than or equal to 0, and equals negative a if a is less than 0
\[ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} \]
the absolute value of a is greater than or equal to 0
\[ |a| \geq 0 \]
the absolute value of negative a equals the absolute value of a
\[ |-a| = |a| \]
the absolute value of a times b equals the absolute value of a times the absolute value of b
\[ |ab| = |a||b| \]
the absolute value of a over b equals the absolute value of a, over the absolute value of b
\[ \left|\dfrac{a}{b}\right| = \dfrac{|a|}{|b|} \]
the absolute value of a plus b is less than or equal to the absolute value of a plus the absolute value of b. This is the Triangle Inequality.
\[ |a+b| \leq |a|+|b| \quad \text{(Triangle Inequality)} \]
Distance Formula
If \(P_1=(x_1,y_1)\) and \(P_2=(x_2,y_2)\) then:
the distance from point 1 to point 2 equals the square root of the quantity x sub 2 minus x sub 1, squared, plus the quantity y sub 2 minus y sub 1, squared
\[ d(P_1,P_2) = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \]
Complex Numbers
i equals the square root of negative 1
\[ i = \sqrt{-1} \]
i squared equals negative 1
\[ i^2 = -1 \]
the square root of negative a equals i times the square root of a, for a greater than or equal to 0
\[ \sqrt{-a} = i\sqrt{a}, \quad a \geq 0 \]
the quantity a plus b i, plus the quantity c plus d i, equals the quantity a plus c, plus the quantity b plus d, times i
\[ (a+bi)+(c+di) = a+c+(b+d)i \]
the quantity a plus b i, minus the quantity c plus d i, equals the quantity a minus c, plus the quantity b minus d, times i
\[ (a+bi)-(c+di) = a-c+(b-d)i \]
the quantity a plus b i, times the quantity c plus d i, equals a times c minus b times d, plus the quantity a times d plus b times c, times i
\[ (a+bi)(c+di) = ac-bd+(ad+bc)i \]
the quantity a plus b i, times the quantity a minus b i, equals a squared plus b squared
\[ (a+bi)(a-bi) = a^2+b^2 \]
the absolute value of a plus b i equals the square root of a squared plus b squared. This is the Complex Modulus.
\[ |a+bi| = \sqrt{a^2+b^2} \quad \text{(Complex Modulus)} \]
the complex conjugate of a plus b i equals a minus b i
\[ \overline{(a+bi)} = a-bi \quad \text{(Complex Conjugate)} \]
the quantity a plus b i, times its conjugate a minus b i, equals the absolute value of a plus b i, squared
\[ (a+bi)\overline{(a+bi)} = |a+bi|^2 \]
Logarithms and Log Properties
Definition:
y equals log base b of x is equivalent to x equals b to the power of y
\[ y = \log_b x \iff x = b^y \]
Special Logarithms:
natural log of x equals log base e of x
\[ \ln x = \log_e x \quad \text{(natural log)} \]
common log of x equals log base 10 of x
\[ \log x = \log_{10} x \quad \text{(common log)} \]
Logarithm Properties:
log base b of b equals 1
\[ \log_b b = 1 \]
log base b of 1 equals 0
\[ \log_b 1 = 0 \]
log base b of b to the power of x, equals x
\[ \log_b b^x = x \]
b raised to the power of log base b of x, equals x
\[ b^{\log_b x} = x \]
log base b of x to the power of r, equals r times log base b of x
\[ \log_b(x^r) = r\log_b x \]
log base b of x times y, equals log base b of x, plus log base b of y
\[ \log_b(xy) = \log_b x + \log_b y \]
log base b of x divided by y, equals log base b of x, minus log base b of y
\[ \log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y \]
Factoring and Solving
Factoring Formulas
x squared minus a squared equals the quantity x plus a, times the quantity x minus a
\[ x^2 - a^2 = (x+a)(x-a) \]
x squared plus 2 a x plus a squared equals the quantity x plus a, squared
\[ x^2 + 2ax + a^2 = (x+a)^2 \]
x squared minus 2 a x plus a squared equals the quantity x minus a, squared
\[ x^2 - 2ax + a^2 = (x-a)^2 \]
x squared plus the quantity a plus b times x, plus a b, equals the quantity x plus a times the quantity x plus b
\[ x^2+(a+b)x+ab = (x+a)(x+b) \]
x cubed plus 3 a x squared plus 3 a squared x plus a cubed equals the quantity x plus a, cubed
\[ x^3+3ax^2+3a^2x+a^3 = (x+a)^3 \]
x cubed minus 3 a x squared plus 3 a squared x minus a cubed equals the quantity x minus a, cubed
\[ x^3-3ax^2+3a^2x-a^3 = (x-a)^3 \]
x cubed plus a cubed equals the quantity x plus a, times the quantity x squared minus a x plus a squared
\[ x^3+a^3 = (x+a)(x^2-ax+a^2) \]
x cubed minus a cubed equals the quantity x minus a, times the quantity x squared plus a x plus a squared
\[ x^3-a^3 = (x-a)(x^2+ax+a^2) \]
x to the power of 2 n minus a to the power of 2 n equals the quantity x to the power of n minus a to the power of n, times the quantity x to the power of n plus a to the power of n
\[ x^{2n}-a^{2n} = (x^n-a^n)(x^n+a^n) \]
x to the power of n minus a to the power of n equals the quantity x minus a, times the quantity x to the power of n minus 1, plus a x to the power of n minus 2, plus dot dot dot, plus a to the power of n minus 1
\[ x^n - a^n = (x-a)(x^{n-1}+ax^{n-2}+\cdots+a^{n-1}) \]
x to the power of n plus a to the power of n equals the quantity x plus a, times the quantity x to the power of n minus 1, minus a x to the power of n minus 2, plus dot dot dot, plus a to the power of n minus 1
\[ x^n + a^n = (x+a)(x^{n-1}-ax^{n-2}+\cdots+a^{n-1}) \]
Quadratic Formula
Solve \(ax^2+bx+c=0,\ a\neq 0\):
x equals negative b plus or minus the square root of b squared minus 4 a c, all over 2 a
\[ x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \]
Functions and Graphs
Constant Function
y equals a, or f of x equals a
\[ y = a \quad \text{or} \quad f(x) = a \]
Graph: horizontal line through \((0,a)\).
Line / Linear Function
y equals m x plus b, or f of x equals m x plus b
\[ y = mx+b \quad \text{or} \quad f(x) = mx+b \]
slope m equals y sub 2 minus y sub 1, over x sub 2 minus x sub 1, which equals rise over run
\[ m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{\text{rise}}{\text{run}} \]
Parabola / Quadratic Function
y equals a times the quantity x minus h squared, plus k. Opens up if a is greater than 0, down if a is less than 0. Vertex at h comma k.
\[ y = a(x-h)^2+k \]
y equals a x squared plus b x plus c. Opens up if a is greater than 0, down if a is less than 0. Vertex at x equals negative b over 2 a.
\[ y = ax^2+bx+c \]
Circle
the quantity x minus h squared, plus the quantity y minus k squared, equals r squared. Centre h comma k, radius r.
\[ (x-h)^2+(y-k)^2 = r^2 \]
Common Algebraic Errors
The left column shows an incorrect statement. The right column explains why it is wrong and gives the correct form.
| Error (incorrect) |
Reason / Correct form / Example |
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2 over 0 is not equal to 0, and 2 over 0 is not equal to 2
\(\dfrac{2}{0} \neq 0\) and \(\dfrac{2}{0} \neq 2\)
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Division by zero is undefined! |
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negative 3 squared is not equal to 9
\(-3^2 \neq 9\)
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negative 3 squared equals negative 9
\(-3^2 = -9\);
the quantity negative 3, squared, equals 9
\((-3)^2 = 9\). Watch parentheses!
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the quantity x squared, cubed, is not equal to x to the fifth
\((x^2)^3 \neq x^5\)
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the quantity x squared, cubed, equals x squared times x squared times x squared, which equals x to the sixth
\((x^2)^3 = x^2\cdot x^2\cdot x^2 = x^6\)
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a over the quantity b plus c is not equal to a over b plus a over c
\(\dfrac{a}{b+c} \neq \dfrac{a}{b}+\dfrac{a}{c}\)
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one half equals 1 over 1 plus 1, which is not equal to 1 over 1 plus 1 over 1, which equals 2
\(\tfrac{1}{2}=\tfrac{1}{1+1}\neq\tfrac{1}{1}+\tfrac{1}{1}=2\)
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1 over x squared plus x cubed is not equal to x to the negative 2 plus x to the negative 3
\(\dfrac{1}{x^2+x^3} \neq x^{-2}+x^{-3}\)
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More complex version of the previous error. |
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the quantity a plus b x, over a, is not equal to 1 plus b x
\(\dfrac{a+bx}{a} \neq 1+bx\)
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the quantity a plus b x, over a, equals a over a plus b x over a, which equals 1 plus b x over a
\(\dfrac{a+bx}{a}=\dfrac{a}{a}+\dfrac{bx}{a}=1+\dfrac{bx}{a}\). Beware incorrect cancelling!
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the quantity x plus a, squared, is not equal to x squared plus a squared
\((x+a)^2 \neq x^2+a^2\)
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the quantity x plus a, squared, equals the quantity x plus a times the quantity x plus a, which equals x squared plus 2 a x plus a squared
\((x+a)^2=(x+a)(x+a)=x^2+2ax+a^2\)
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the square root of x squared plus a squared is not equal to x plus a
\(\sqrt{x^2+a^2} \neq x+a\)
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5 equals the square root of 25 equals the square root of 3 squared plus 4 squared, which is not equal to the square root of 3 squared plus the square root of 4 squared, which equals 3 plus 4 equals 7
\(5=\sqrt{25}=\sqrt{3^2+4^2}\neq\sqrt{3^2}+\sqrt{4^2}=3+4=7\)
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the square root of negative x squared plus a squared is not equal to negative the square root of x squared plus a squared
\(\sqrt{-x^2+a^2} \neq -\sqrt{x^2+a^2}\)
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the square root of negative x squared plus a squared equals the quantity negative x squared plus a squared, to the one half power
\(\sqrt{-x^2+a^2}=(-x^2+a^2)^{1/2}\)
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a times the fraction b over c is not equal to a b over c
\(a\!\left(\dfrac{b}{c}\right) \neq \dfrac{ab}{c}\)
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a times b over c equals a over 1 times b over c equals a over 1 times c over b equals a c over b
\(a\!\left(\dfrac{b}{c}\right)=\dfrac{a}{1}\cdot\dfrac{b}{c}=\dfrac{a}{1}\cdot\dfrac{c}{b}=\dfrac{ac}{b}\)
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the fraction a over b, divided by c, is not equal to a c over b
\(\dfrac{\,\dfrac{a}{b}\,}{c} \neq \dfrac{ac}{b}\)
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the fraction a over b, divided by c, equals a over b times 1 over c, which equals a over b c
\(\dfrac{\,\dfrac{a}{b}\,}{c}=\dfrac{a}{b}\cdot\dfrac{1}{c}=\dfrac{a}{bc}\)
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