GCSE(H),

Simultaneous equations can also be solved when one of the equations is in a quadratic form. In this instance:

• Make the subject of the linear equation the same as the power term in the quadratic equation;

• Substitute into the quadratic equation;

• Solve the quadratic equation, using factorisation, completing the square, or the quadratic formula;

• Solve the second variable by substitution into the linear equation;

Examples

1. Solve y=2x+10 text( and ) y=-x^2-4x+5

Answer: x=-1, y=8 text( and ) x=-5, y=0

Substitute the linear equation into the quadratic equation using the variable y:

2x+10=-x^2-4x+5

x^2+4x+2x+10-5=0

x^2+6x+5=0

Factorise to (x+1)(x+5)=0

With x=-1, y=8 text( and ) x=-5, y=0

Check: 8 = -(-1)2 - 4(-1) + 5 (true) and 0 = -(-5)2 - 4(-5) + 5 (true)

2. Solve x-2y=4 text( and ) 2y=x^2-8x+4.

Answer: x=1, y=-1frac(1)(2) text( and ) x=8, y=2

Rearrange x-2y=4 text( to ) 2y=x-4

Substitute for the linear equation; x-4=x^2-8x+4

Rearrange and set one side to zero: x^2-8x-x-4-4=0

Factorise x^2 - 9x - 8 to 0=(x-1)(x-8)

Substitute into the linear equation to get y = -1frac(1)(2), y=2

Check: -1frac(1)(2) = 0.5(1)^2 - 4(1) + 2 (true) and (2) = 0.5(8)^2 - 4(8) + 2 (true)