IEB Grade 12 Algebra, Equations Cheat Sheet

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Everything you need for the algebra question in Paper 1 (about 25 marks), on two printable pages.

Quadratic equations

Standard form first: $a x^{2} + b x + c = 0$ . Never divide by $x$ ; factorise it out.
Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$
k-substitution: if an expression repeats, let $k$ equal it, solve for $k$ , then solve for $x$ .
Fractions: state restrictions (denominators $\neq = 0$ ), multiply every term by the LCD, check answers against the restrictions.

Isolate the surd, square both sides, solve, then check every answer in the original equation.
Quick filter: $\dots \geq 0$ , so the other side of the isolated equation must be $\geq 0$ .

Get 0 on one side, factorise, find critical values.
$a > 0$ : ' $< 0$ ' lies between the roots, ' $> 0$ ' lies outside them.
$(x - p)^{2} < 0$ has no real solution; $(x - p)^{2} \geq 0$ is true for all real $x$ .

Make $x$ or $y$ the subject of the linear equation, substitute into the quadratic, solve, then substitute back into the linear one.
Each solution pair is a point of intersection of the two graphs.

$Δ < 0$ : non-real
$Δ = 0$ : real, equal, rational
$Δ > 0$ and a perfect square: real, unequal, rational
$Δ > 0$ and not a perfect square: real, unequal, irrational
Line meets curve: substitute, then $Δ > 0$ (cuts twice), $Δ = 0$ (tangent), $Δ < 0$ (misses).

For $a > 0$ : $a^{m} a^{n} = a^{m + n}$ , $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ , $(a^{m})^{n} = a^{mn}$ , $a^{- n} = \frac{1}{a ^{n}}$ , $a^{\frac{m}{n}} = n a^{m}$ . The first four also hold for any $a \neq = 0$ when $m$ and $n$ are integers.
Same base: $a^{x} = a^{y} \Rightarrow x = y$ (for $a > 0$ , $a \neq = 1$ ). Common factor: $3^{x + 1} + 3^{x} = 3^{x} (3 + 1)$ .
$2^{x} > 0$ always, so reject solutions like $2^{x} = - 4$ .
$ab = a b$ but $a + b \neq = a + b$ . Add only like surds.