Solving Quadratics by Factorising
How do I solve a quadratic equation using factorisation?
- Rearrange it into the form ax2 + bx + c = 0
- zero must be on one side
- it is easier to use the side where a is positive
- zero must be on one side
- Factorise the quadratic and solve each bracket equal to zero
- If (x + 4)(x - 1) = 0, then either x + 4 = 0 or x - 1 = 0
- Because if A × B = 0, then either A = 0 or B = 0
- If (x + 4)(x - 1) = 0, then either x + 4 = 0 or x - 1 = 0
- To solve
- …solve “first bracket = 0”:
- x – 3 = 0
- add 3 to both sides: x = 3
- …and solve “second bracket = 0”
- x + 7 = 0
- subtract 7 from both sides: x = -7
- The two solutions are x = 3 or x = -7
- The solutions have the opposite signs to the numbers in the brackets
- …solve “first bracket = 0”:
- To solve
- …solve “first bracket = 0”
- 2x – 3 = 0
- add 3 to both sides: 2x = 3
- divide both sides by 2: x =
- …solve “second bracket = 0”
- 3x + 5 = 0
- subtract 5 from both sides: 3x = -5
- divide both sides by 3: x =
- The two solutions are x = or x =
- …solve “first bracket = 0”
- To solve
- it may help to think of x as (x – 0) or (x)
- …solve “first bracket = 0”
- (x) = 0, so x = 0
- …solve “second bracket = 0”
- x – 4 = 0
- add 4 to both sides: x = 4
- The two solutions are x = 0 or x = 4
- It is a common mistake to divide both sides by x at the beginning - you will lose a solution (the x = 0 solution)
Examiner Tip
- Use a calculator to check your final solutions!
- Calculators also help you to factorise (if you're struggling with that step)
- A calculator gives solutions to as x = and x =
- "Reverse" the method above to factorise!
- Warning: a calculator gives solutions to 12x2 + 2x – 4 = 0 as x = and x =
- But 12x2 + 2x – 4 ≠ as these brackets expand to 6x2 + ... not 12x2 + ...
- Multiply by 2 to correct this
- 12x2 + 2x – 4 =
Worked example
Solve
Set the first bracket equal to zero
x – 2 = 0
Add 2 to both sides
x = 2
Set the second bracket equal to zero
x + 5 = 0
Subtract 5 from both sides
x = -5
Write both solutions together using “or”
x = 2 or x = -5
Set the first bracket equal to zero
8x + 7 = 0
Subtract 7 from both sides
8x = -7
Divide both sides by 8
x =
Set the second bracket equal to zero
2x - 3 = 0
Add 3 to both sides
2x = 3
Divide both sides by 2
x =
Write both solutions together using “or”
x = or x =
Solve
Do not divide both sides by x (this will lose a solution at the end)
Set the first “bracket” equal to zero
(x) = 0
Solve this equation to find x
x = 0
Set the second bracket equal to zero
5x - 1 = 0
Add 1 to both sides
5x = 1
Divide both sides by 5
x =
Write both solutions together using “or”
x = 0 or x =