Quadratic Equations
A quadratic equation is an equation where the highest power of the variable is squared. In GCSE Maths, many quadratic equations can be solved by factorising.
A quadratic equation often looks like:
x² + bx + c = 0
Solving quadratics by factorising
To solve a quadratic by factorising, put the equation equal to zero, factorise, then set each bracket equal to zero.
Example 1: Solve x² + 5x + 6 = 0
Factorise the quadratic:
x² + 5x + 6 = (x + 2)(x + 3)
So:
(x + 2)(x + 3) = 0
Set each bracket equal to zero:
x + 2 = 0 or x + 3 = 0
x = -2 or x = -3
Why each bracket equals zero
If two things multiply to make zero, at least one of them must be zero. This is why we set each bracket equal to zero.
Example 2: Solve x² - 7x + 10 = 0
Factorise:
x² - 7x + 10 = (x - 5)(x - 2)
Set each bracket equal to zero:
x - 5 = 0 or x - 2 = 0
x = 5 or x = 2
Quadratics with a negative constant
If the constant is negative, one bracket usually has a plus sign and the other has a minus sign.
Example 3: Solve x² + x - 12 = 0
We need two numbers that multiply to -12 and add to 1.
Those numbers are 4 and -3.
x² + x - 12 = (x + 4)(x - 3)
Set each bracket equal to zero:
x + 4 = 0 or x - 3 = 0
x = -4 or x = 3
Rearranging before solving
Sometimes the equation is not already equal to zero. Rearrange it first.
Example 4: Solve x² + 6x = 16
Move 16 to the left side:
x² + 6x - 16 = 0
Factorise:
(x + 8)(x - 2) = 0
x = -8 or x = 2
Checking your answers
You can check a solution by substituting it back into the original equation.
Example 5: Check a solution
Check x = 2 in x² + 6x = 16.
2² + 6(2) = 4 + 12 = 16
Since the left side equals 16, x = 2 is correct.
A common mistake is factorising correctly but forgetting to set each bracket equal to zero. The factorised form is not the final answer; you must find the values of x.
Always check whether the equation is equal to zero before factorising. If it is not, rearrange it first.
Video explanation
A short Worthing Maths Tutor video explanation for quadratic equations can be embedded here later to improve student engagement and time on page.
Practice questions
- Solve x² + 7x + 12 = 0.
- Solve x² - 9x + 20 = 0.
- Solve x² + 2x - 15 = 0.
- Solve x² - 4x = 21.
- Check whether x = 3 is a solution of x² + x - 12 = 0.
Answers
- x = -3 or x = -4
- x = 5 or x = 4
- x = 3 or x = -5
- x = 7 or x = -3
- Yes, because 3² + 3 - 12 = 0
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