Equations and InequalitiesAQA GCSE Maths: Revision notes

A linear equation has an unknown raised only to the power 1. To solve one algebraically:

1. Expand any brackets
2. Collect the unknown terms on one side and number terms on the other
3. Simplify each side
4. Divide to leave the unknown on its own

This works even when the unknown appears on both sides of the equation.

If an equation is too complex to solve exactly, you can find an approximate solution using a graph — plot both sides as separate lines and read off the $x$-value where they intersect.

A quadratic equation contains a squared term ($x^2$). There are three algebraic methods:

• Factorising: rewrite as two brackets multiplying to zero, then solve each bracket $=0$
• Completing the square: rewrite in the form $(x+p)^2 + q = 0$ and rearrange
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ for $ax^2+bx+c=0$

Some quadratics need rearranging into the form $ax^2 + bx + c = 0$ before you can solve them. As with linear equations, approximate solutions can also be found by reading the roots off a plotted graph.

Simultaneous equations are two equations that share the same two unknowns, solved together so both are true at once.

• Linear/linear: solve algebraically by elimination or substitution
• Linear/quadratic (Higher): solve by substitution — substitute the linear equation into the quadratic one
• Solutions can also be found approximately from a graph, by reading the coordinates of the point(s) where the two graphs intersect

Real problems (including geometry problems) can be translated into an equation or a pair of simultaneous equations, then solved and the solution interpreted in context.

Some equations cannot be solved exactly with algebra. For these, iteration is used: a recursive formula such as $x_{n+1} = f(x_n)$ is applied repeatedly, starting from a given value, so that the answer converges towards an approximate solution.

An inequality compares expressions using $<$, $>$, $\leq$ or $\geq$ rather than $=$.

• Linear inequalities in one variable are solved much like equations, but flip the inequality sign if you multiply or divide by a negative number
• Higher tier also covers linear inequalities in two variables and quadratic inequalities in one variable
• Solutions can be shown on a number line, in set notation, or on a graph

On a number line, use an open circle for a strict inequality ($<$ or $>$, boundary not included) and a closed circle for $\leq$ or $\geq$ (boundary included). On a graph, a dashed line shows a strict inequality boundary and a solid line shows an included boundary.