Quadratic Equation Methods
If you have to solve a quadratic equation but are not told which method to use, here is a guide as to what to do
When should I solve by factorisation?
- When the question asks to solve by factorisation
- For example, part (a) Factorise 6x2 + 7x – 3, part (b) Solve 6x2 + 7x – 3 = 0
- When solving two-term quadratic equations
- For example, solve x2 – 4x = 0
- …by taking out a common factor of x to get x(x – 4) = 0
- ...giving x = 0 and x = 4
- For example, solve x2 – 9 = 0
- …using the difference of two squares to factorise it as (x + 3)(x – 3) = 0
- ...giving x = -3 and x = 3
- (Or by rearranging to x2 = 9 and using ±√ to get x = = ±3)
- For example, solve x2 – 4x = 0
When should I use the quadratic formula?
- When the question says to leave solutions correct to a given accuracy (2 decimal places, 3 significant figures etc)
- When the quadratic formula may be faster than factorising
- It's quicker to solve 36x2 + 33x – 20 = 0 using the quadratic formula then by factorisation
- If in doubt, use the quadratic formula - it always works
When should I solve by completing the square?
- When part (a) of a question says to complete the square and part (b) says to use part (a) to solve the equation
- When making x the subject of harder formulae containing x2 and x terms
- For example, make x the subject of the formula x2 + 6x = y
- Complete the square: (x + 3)2 – 9 = y
- Add 9 to both sides: (x + 3)2 = y + 9
- Take square roots and use ±:
- Subtract 3:
- For example, make x the subject of the formula x2 + 6x = y
Examiner Tip
- Calculators can solve quadratic equations so use them to check your solutions
- If the solutions on your calculator are whole numbers or fractions (with no square roots), this means the quadratic equation does factorise
Worked example
“Correct to 2 decimal places” suggests using the quadratic formula
Substitute a = 1, b = -7 and c = 2 into the formula, putting brackets around any negative numbers
Use a calculator to find each solution
x = 6.70156… or 0.2984...
Round your final answers to 2 decimal places
x = 6.70 or x = 0.30
Method 1
If you cannot spot the factorisation, use the quadratic formula
Substitute a = 16, b = -82 and c = 45 into the formula, putting brackets around any negative numbers
Use a calculator to find each solution
x = or x =
Method 2
If you do spot the factorisation, (2x – 9)(8x – 5), then use that method instead
Set the first bracket equal to zero
Add 9 to both sides then divide by 2
Set the second bracket equal to zero
Add 5 to both sides then divide by 8
x = or x =
This question wants you to complete the square first
Find p (by halving the middle number)
Write x2 + 6x as (x + p)2 - p2
Replace x2 + 6x with (x + 3)2 – 9 in the equation
Make x the subject of the equation (start by adding 4 to both sides)
Take square roots of both sides (include a ± sign to get both solutions)
Subtract 3 from both sides
Find each solution separately using + first, then - second
x = - 5, x = - 1
Even though the quadratic factorises to (x + 5)(x + 1), this is not the method asked for in the question