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 possible, factorising is usually the easiest way to solve a quadratic equation
- Even on the calculator paper, if you can spot a factorisation quickly, use this approach
When should I use the quadratic formula?
- If the coefficients (a, b and c) are large, factorising and completing the square can be difficult or slow
- The quadratic formula lends itself to using a calculator
- Some modern calculators will solve quadratic equations directly, with no need to use the formula
- Typically the quadratic formula would be used when rounding is involved
- For example, if a question says to leave solutions correct to 2 decimal places or 3 significant figures
- However, the quadratic formula is also useful when answers need to be exact
- The formula lends itself to surd form after simplifying some of the values within it
- e.g.
- If in doubt, use the quadratic formula - it always works
When should I solve by completing the square?
- A question may direct you to solve by completing the square
- e.g. Part (a) says to complete the square and part (b) says 'hence' or 'use part (a)' to solve ...
- Completing the square may have already happened for other reasons
- e.g. Completing the square allows the coordinates of the turning point on a quadratic graph to be found easily
- If this has been done in an earlier part of a question, use it to solve the quadratic equation
Examiner Tip
- Calculators can solve quadratic equations
- Double check you've entered the equation correctly, in the correct format
- Use this feature to check your answers where possible
- 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 and/or a calculator
For accuracy, it is a good idea to use both - use the formula and calculator as normal first
Then use the quadratic solver feature to check your solutions
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
If your calculator has a quadratic equation solver, use it to check your answers
Method 1
The coefficients are large and so the factorisation, even if possible, is hard to spot
Therefore, one method to use is the quadratic formula - it always works!
The solution below is the manual way to use a calculator, but as above, if your calculator has a quadratic solver feature, you may use that
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 persevere with the factorisation 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 =
Notice this question does not use the phrase 'completing the square' but shows the form of it instead
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