Simultaneous equations with linear and quadratic - Solving simultaneous equations - AQA - GCSE Maths Revision - AQA

Simultaneous equations with one linear and one quadratic - Higher

A does not contain any powers higher than 1.

A is an equation with the highest power of 2. For example:

\(y = x + 3\) is a linear equation and \(y = x^2 + 3x\) is a quadratic equation

\(y = x + 3\)

\(y = x^2 + 3x\)

Substitute \(y = x + 3\) into the quadratic equation to create an equation which can be factorised and solved.

\(\mathbf{x~+~3} = x^2 + 3x\)

Rearrange the equation to get all terms on one side, so subtract \(x\) and \(-3\) from both sides:

\(x + 3 - x - 3 = x^2 + 3x - x - 3\)

\(0 = x^2 + 2x - 3\)

Factorise this equation:

\((x+3)(x-1) = 0\)

If the product of two brackets is zero, then one or both brackets must also be equal to zero.

To solve, put each bracket equal to zero.

\(\begin{array}{rcl} x + 3 & = & 0 \\ -3 && -3 \\ x & = & -3 \end{array}\)

\(\begin{array}{rcl} x - 1 & = & 0 \\ +1 && +1 \\ x & = & 1 \end{array}\)

To find the values for \(y\), substitute the two values for \(x\) into the original linear equation.

\(y = x + 3\) when \(x = -3\)

\(y = \mathbf{-3} + 3\)

\(y = 0\)

\(y = x + 3\) when \(x = 1\)

\(y = \mathbf{1} + 3\)

\(y = 4\)

The answers are now in pairs: when \(x = -3\), \(y = 0\) and when \(x = 1\), \(y = 4\)