Before You Begin
Concept Review
Tap any topic to review key formulas and a worked example.
Quadratics & Parabolas
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Key Formulas to Memorize
Standard: $ax^2+bx+c=0$
Quadratic Formula: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$
Discriminant: $\Delta = b^2-4ac$
Vertex form: $f(x)=a(x-h)^2+k$, vertex $= (h,k)$
Quadratic Formula: $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$
Discriminant: $\Delta = b^2-4ac$
Vertex form: $f(x)=a(x-h)^2+k$, vertex $= (h,k)$
When $\Delta>0$: 2 real roots · $\Delta=0$: 1 repeated root · $\Delta<0$: no real roots
Worked Example
Solve $x^2-5x+6=0$.
$\Delta = 25-24=1$. So $x = \dfrac{5\pm1}{2}$, giving $x=3$ or $x=2$.
$\Delta = 25-24=1$. So $x = \dfrac{5\pm1}{2}$, giving $x=3$ or $x=2$.
Polynomials & Factoring
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Key Rules
Factor Theorem: $(x-a)$ is a factor of $f(x)$ iff $f(a)=0$
Difference of cubes: $a^3-b^3=(a-b)(a^2+ab+b^2)$
Rational simplification: cancel common factors, state restrictions
Difference of cubes: $a^3-b^3=(a-b)(a^2+ab+b^2)$
Rational simplification: cancel common factors, state restrictions
Worked Example
Factor $x^3-8$. Using difference of cubes with $a=x,\,b=2$:
$x^3-8=(x-2)(x^2+2x+4)$.
$x^3-8=(x-2)(x^2+2x+4)$.
Functions & Inverses
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Key Rules
To find $f^{-1}$: replace $f(x)$ with $y$, swap $x$ and $y$, solve for $y$.
Verify: $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$
Verify: $f(f^{-1}(x))=x$ and $f^{-1}(f(x))=x$
Worked Example
$f(x)=3x-5$. Let $y=3x-5 \Rightarrow x=\dfrac{y+5}{3}$. So $f^{-1}(x)=\dfrac{x+5}{3}$.
Exponential & Logarithms
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Essential Laws
$b^x=N \Leftrightarrow \log_b N = x$
Product: $\log(mn)=\log m+\log n$
Quotient: $\log\!\left(\tfrac{m}{n}\right)=\log m-\log n$
Power: $\log(m^n)=n\log m$
Change of base: $\log_b a = \dfrac{\ln a}{\ln b}$
Product: $\log(mn)=\log m+\log n$
Quotient: $\log\!\left(\tfrac{m}{n}\right)=\log m-\log n$
Power: $\log(m^n)=n\log m$
Change of base: $\log_b a = \dfrac{\ln a}{\ln b}$
Worked Example
Solve $2^{x+1}=32$. Since $32=2^5$: $x+1=5 \Rightarrow x=4$.
Sequences & Series
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Formulas
Arithmetic: $a_n=a_1+(n-1)d$
Geometric: $a_n=a_1 \cdot r^{n-1}$
Geo partial sum: $S_n=a_1\cdot\dfrac{1-r^n}{1-r}$
Geometric: $a_n=a_1 \cdot r^{n-1}$
Geo partial sum: $S_n=a_1\cdot\dfrac{1-r^n}{1-r}$
Worked Example
Arithmetic: $a_1=3,\,d=4$. Find $a_{10}$.
$a_{10}=3+(10-1)(4)=3+36=39$.
$a_{10}=3+(10-1)(4)=3+36=39$.
Systems & Matrices
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Matrix Multiplication 2×2
$\begin{pmatrix}a&b\\c&d\end{pmatrix}\begin{pmatrix}e&f\\g&h\end{pmatrix}=\begin{pmatrix}ae+bg&af+bh\\ce+dg&cf+dh\end{pmatrix}$
Worked Example
Solve: $x+y=7,\;2x-y=2$. Add both equations: $3x=9 \Rightarrow x=3,\;y=4$.
Inequalities, Complex, Radicals
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Key Rules
$|ax+b|<c \Rightarrow -c<ax+b<c$
$i^2=-1$; $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Radical: square both sides, check for extraneous solutions
$i^2=-1$; $(a+bi)(c+di)=(ac-bd)+(ad+bc)i$
Radical: square both sides, check for extraneous solutions
Worked Example
$(3+2i)(1-4i)=3-12i+2i-8i^2=3-10i+8=11-10i$.
Conics & Binomial Theorem
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Key Formulas
Circle: $(x-h)^2+(y-k)^2=r^2$, center $(h,k)$, radius $r$
Binomial: $(x+a)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}a^k$
Binomial: $(x+a)^n = \sum_{k=0}^{n}\binom{n}{k}x^{n-k}a^k$
Worked Example
Coefficient of $x^2$ in $(x+2)^5$: term is $\binom{5}{3}x^2\cdot 2^3=10\cdot8=80$.
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