Simultaneous Equations Linear/Quadratic

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

• Only one of the variables will be a squared term: replace the other one;

• 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;

• Check answers by substitution.

Remember there will probably be two answers (although there may be one (twice) or none. For each answer, find the corresponding unknown of the other variable.

Example 1

Solve `y=2x+10` and `y=-x^2-4x+5`

Replace the `y` in the quadratic with `2x + 10` from the linear equation and rearrange to have zero on one side:

Replace `y`	``	`2x`	`+10`	`=`	`-x^2`	`-4x`	`+5`
Add `x^2` both sides	`x^2`	`+2x`	`+10`	`=`	``	`-4x`	`+5`
Add `4x` both sides	`x^2`	`+6x`	`+10`	`=`	``	``	`+5`
Subtract 5 both sides	`x^2`	`+6x`	`+5`	`=`	`0`	``	``

Factorise the quadratic: `(x+1)(x+5)=0`

Solutions for `x` are -1 and -5, as that is what makes each bracket equal to zero

using the linear equation:

When `x=-1` then `y = 2(-1)+10=8`

When `x=-5` then `y = 2(-5)+ 10 = 0`

Solutions are `x=-1, y=8` and `x=-5, y=0`

Check:

8 = -(-1)² - 4(-1) + 5 ✔

0 = -(-5)² - 4(-5) + 5 ✔

Answer: `x=-1, y=8` and `x=-5, y=0`

Example 2

Solve `x-2y=4` and `2y=x^2-8x+4`.

Rearrange `x-2y=4` to make `2y` the subject:

`2y=x-4`

Using the quadratic:		`2y`	`=`	`x^2`	`-8x`	`+4`
Replace `2y`:	`x`	`-4`	`=`	`x^2`	`-8x`	`+4`
Subtract `x` from both sides:	``	`-4`	`=`	`x^2`	`-9x`	`+4`
Add 4 to both sides:	``	`0`	`=`	`x^2`	`-9x`	`+8`

Factorise `x^2-9x+8` to `(x-1)(x-8)`

Gives solutions of `x=1` and `x=8` as this makes each bracket zero

Substitute into the linear equations to get `y` for each value

when `x=1`, then `(1)-2y=4` and `y=-frac(3)(2)`

when `x=8`, then `(8)-2y=4` and `y=-2`

Solutions are: `x=1`, `y=-frac(3)(2)`

and: `x=8`, `y=2`

Check:

`2(-frac(3)(2))=(1)^2-8(1)+4` ✔

`2(2)=(8)^2-8(8)+4` ✔

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