IEB Grade 11 Algebra, Equations and Inequalities Cheat Sheet

Algebra is the biggest part of Grade 11 Paper 1 (about 45 marks). Every method is on these two pages.

Exponents and surds

• $a^m{a}^n=a^{m+n}$, $\frac{a^m}{a^n}=a^{m-n}$, $(a^m{)}^n=a^{mn}$, $a^{-n}=\frac{1}{a^n}$, $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

• Write numbers as prime powers: $16=2^4$, $27=3^3$.

• Simplify surds: $\sqrt{75}=5\sqrt{3}$. Add only like surds.

• Rationalise: multiply by the surd, or by the conjugate for two terms: $\frac{6}{\sqrt{5}-\sqrt{2}}\cdot \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$.

Exponential and surd equations

• Same base: $a^x=a^y\Rightarrow x=y$. Common factor: $5^{x+1}-5^x=5^x(5-1)$.

• k-substitution: $4^x=(2^x{)}^2$; reject $2^x\le 0$.

• $x^{\frac{2}{3}}=9\Rightarrow x=\pm 27$ (even numerator gives $\pm$).

• Surd equations: isolate, square, solve, check every answer.

Quadratic equations

• Standard form, then factorise, complete the square, or use $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

• Completing the square: divide by $a$, add ${\left(\frac{b}{2}\right)}^2$ to both sides, take $\pm \sqrt{}$.

• Turning point form: $y=a(x+p{)}^2+q$ gives $(-p;q)$.

• Fractions: restrictions first, multiply every term by the LCD, check.

Inequalities

• Factorise, find critical values, then sketch or use a number line.

• $a>0$: '$<0$' between roots, '$>0$' outside. Multiply by a negative: reverse the sign.

• Interval notation: $(-2;3]$; infinity always takes a round bracket.

Simultaneous equations and nature of roots

• Substitute the linear equation into the quadratic; substitute back into the linear one.

• $\Delta =b^2-4ac$: $<0$ non-real; $=0$ equal; $>0$ perfect square: rational; $>0$ not a square: irrational.

• Real roots: $\Delta \ge 0$. If $k$ is the coefficient of $x^2$, also $k\ne 0$.

Know by heart

• Exponent laws

• Completing the square and the derivation of the quadratic formula

• The nature of roots cases

On the formula sheet

• The quadratic formula

Common traps

• Dividing by $x$ and losing $x=0$.

• Adding unlike surds.

• Not checking surd equation answers.

• Dividing an inequality by an expression with $x$ in it.