Simultaneous equations with one linear and one quadratic - Higher

A linear equation does not contain any powers higher than 1. A quadratic equation contains a variable that's highest power is 2. For example:

y = x + 3 is a linear equation and y = x^2 + 3x is a quadratic equation.

Solving simultaneous equations with one linear and one quadratic

Algebraic skills of substitution and factorising are required to solve these equations.

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When solving simultaneous equations with a linear and quadratic equation, there will usually be two pairs of answers.

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.

y = x^2 + 3x

Substitute y = x + 3:

\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

0 = x^2 + 2x - 3

Factorise this equation:

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

If the product of two numbers is zero, then one or both numbers 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