IEB · Grade 12 · Mathematics Paper 1 · Algebra, equations and inequalities
In the print dialog, choose Save as PDF. IEB Grade 12 Algebra, Equations and Inequalities Cheat Sheet 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. Surd equations 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 $\geq 0$ . Quadratic inequalities 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 $x$ . Simultaneous equations 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. Nature of roots: $Δ = b^{2} - 4 a c$ (with $a$ , $b$ , $c$ rational) $Δ < 0$ : 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). Exponents and surds 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. Know by heart Exponent laws and how to write numbers as prime powers The four cases for the nature of roots The steps for surd equations, including the check On the formula sheet The quadratic formula Common traps Dividing by $x$ and losing $x = 0$ . $(x - 1)^{2} \neq = x^{2} - 1$ when squaring a surd equation. Writing 'or' inequality answers as one combined inequality. Substituting back into the quadratic instead of the linear equation.
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IEB Grade 12 Algebra, Equations and Inequalities Cheat Sheet

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.

Surd equations

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 $\geq 0$ .

Quadratic inequalities

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 $x$ .

Simultaneous equations

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.

Nature of roots: $Δ = b^{2} - 4 a c$ (with $a$ , $b$ , $c$ rational)

$Δ < 0$ : 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).

Exponents and surds

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.

Know by heart

Exponent laws and how to write numbers as prime powers
The four cases for the nature of roots
The steps for surd equations, including the check

On the formula sheet

The quadratic formula

Common traps

Dividing by $x$ and losing $x = 0$ .
$(x - 1)^{2} \neq = x^{2} - 1$ when squaring a surd equation.
Writing 'or' inequality answers as one combined inequality.
Substituting back into the quadratic instead of the linear equation.

IEB Grade 12 Algebra, Equations and Inequalities Cheat Sheet

Quadratic equations

Surd equations

Quadratic inequalities

Simultaneous equations

Nature of roots: Δ=b2−4ac (with a, b, c rational)

Exponents and surds

Know by heart

On the formula sheet

Common traps

Nature of roots: $Δ = b^{2} - 4 a c$ (with $a$ , $b$ , $c$ rational)