Solving Quadratic Equations
How do I solve a quadratic equation using factorisation?
- Factorise the quadratic and solve each bracket equal to zero
- To solve
- solve 2x – 3 = 0 to get x =
- solve 3x + 5 = 0 to get x =
- To solve don't forget to solve x = 0
- The two solutions are x = 0 or x = 4
- It is a common mistake to divide by x at the beginning (you will lose a solution)
- The two solutions are x = 0 or x = 4
How do I solve a quadratic equation by completing the square?
- To solve x2 + bx + c = 0
- replace the first two terms, x2 + bx, with (x + p)2 - p2 where p is half of b
- this is called completing the square
- x2 + bx + c = 0 becomes
- (x + p)2 - p2 + c = 0 where p is half of b
- x2 + bx + c = 0 becomes
- rearrange this equation to make x the subject (using ±√)
- For example, solve x2 + 10x + 9 = 0 by completing the square
- x2 + 10x becomes (x + 5)2 - 52
- so x2 + 10x + 9 = 0 becomes (x + 5)2 - 52 + 9 = 0
- make x the subject (using ±√)
- (x + 5)2 - 25 + 9 = 0
- (x + 5)2 = 16
- x + 5 = ±√16
- x = ±4 - 5
- x = -1 or x = -9
- If the equation is ax2 + bx + c = 0 with a number in front of x2, then divide both sides by a first, before completing the square
How do I use the quadratic formula to solve a quadratic equation?
- The quadratic formula is
- Read off the values of a, b and c from the equation
- Substitute these into the formula
- Write this line of working in the exam
- Put brackets around any negative numbers being substituted in
- To solve 2x2 - 7x - 3 = 0 using the quadratic formula:
- a = 2, b = -7 and c = -3
- Type this into a calculator
- once with + for ± and once with - for ±
- The solutions are x = 3.886 and x = -0.386 (to 3 dp)
- Rounding is often asked for in the question
- The calculator also gives these solutions in exact form (surd form)
-
- x = and x =
- You need to be able to find solutions in exact / surd form without a calculator
- this means working out (-7)2 - 4 × 2 × (-3)
-
What is the discriminant?
- The part of the formula under the square root (b2 – 4ac) is called the discriminant
- The sign of this value tells you if there are 0, 1 or 2 solutions
- If b2 – 4ac > 0 (positive)
- then there are 2 different solutions
- If b2 – 4ac = 0 (zero)
- then there is only 1 solution
- sometimes called "repeated solutions"
- If b2 – 4ac < 0 (negative)
- then there are no solutions
- If your calculator gives you solutions with i terms in, these are "complex" and not what we are looking for
- Interestingly, if b2 – 4ac is a perfect square number ( 1, 4, 9, 16, …) then the quadratic expression could have been factorised!
- If b2 – 4ac > 0 (positive)
How does completing the square link to the quadratic formula?
- The quadratic formula actually comes from completing the square of ax2 + bx + c = 0
- You can see hints of this when you solve quadratics
- For example, solving x2 + 10x + 9 = 0
- by completing the square, (x + 5)2 = 16 so x = -5 ± 4
- by the quadratic formula, = -5 ± 4
- For example, solving x2 + 10x + 9 = 0
Can I use my calculator to solve quadratic equations?
- Yes, in the calculator paper, 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
- the right-hand side expands to 6x2 + ... ,not 12x2 + ...
- Correct this by multiplying the right by 2
- 12x2 + 2x – 4
- "Reverse" the method above to factorise
Exam Tip
- Make sure the quadratic equation has "= 0" on the right-hand side, otherwise it needs rearranging first
- rearrange to have ax2 on its positive side (a>0)
- Always look for how the question wants you to leave your final answers
- 2 decimal places, 3 significant figures, in exact form, etc
Worked example
(a)
Solve , giving your answers in exact form.
“exact form” suggests using the quadratic formula (surds will be in the answer)
Substitute a = 1, b = -7 and c = 2 into the formula, putting brackets around any negative numbers
Work out (-7)2 - 4 × 1 × 2 and simplify
This is as simplified as possible
(b)
Solve
Method 1
If you cannot spot the factorisation and this is in the calculator paper, 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
or
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
or
(c)
By writing in the form , solve
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
Even though the quadratic factorises to (x + 5)(x + 1), this is not the method asked for in the question