Maths
GCSE
A18 Solving Quadratic Equations
Solving Equations and Inequalities
Algebra
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Contents
Solve Quadratic Equations by factorising ... 3 Solve Quadratic Equations by completing the square ... 11 Solve Quadratic Equations by using the quadratic formula... 16
Information
Solve Quadratic Equations by factorising
A quadratic equation is an equation where the highest power of x is x2. The standard format for a quadratic equation is
ax2 + bx + c = 0
To solve a quadratic equation, (finding values of x that make the equation true), you usually need to factorise the equation first and then solve the resulting equation.
Factorising is the opposite of expanding brackets.
Factorising a quadratic equation means putting the equation into 2 brackets.
To solve a quadratic equation Step 1:
Arrange the equation into the standard format if it isn't already in this format.
ax2 + bx + c = 0 Step 2:
Look for 2 numbers that when multiplied give you the ‘c’ term but also when added or subtracted together, give you the ‘b’ term (the coefficient of x).
Step 3:
Put these numbers into 2 brackets and check the +/- signs.
Step 4:
Solve the resulting equation.
Step 5:
Check your answers by substituting in the original quadratic equation.
Example
Solve x2 + 3x = 10 by factorising.
Step 1
Arrange the equation into the standard format.
x2 + 3x = 10 when rearranged becomes x2 + 3x – 10 = 0.
Step 2
Look for 2 numbers when multiplied together give you 10 and when added/subtracted together give you 3.
1 x 10 = 10 when +/- give 11 or 9 2 x 5 = 10 when +/- give 7 or 3
2 x 5 = 10 and 5 – 2 = 3 so put these 2 numbers into 2 brackets.
Step 3
(x 2)(x 5) = 0
Fill in the signs so that the 2 and 5 will give +3 ( the coefficient of the x term ‘b’) and -10 (for term ‘c’).
It must be -2 and +5 to get +3 and -2 x 5 = -10 (x – 2)(x + 5) = 0
Check your answer by expanding the brackets out to get the original equation.
(x – 2)(x + 5)
= x2 + 5x – 2x – 10 = x2 + 3x – 10 ✔
Step 4
Then solve the equation.
(x – 2)(x + 5) = 0.
(x – 2) = 0 or (x + 5) = 0 x – 2 = 0
x = 2 or x + 5 = 0 x = -5
The solutions are x = 2 or x = -5
Step 5
Check your answers by substituting values for x into the original equation.
x2 + 3x = 10 when x = 2 4 + 6 = 10 ✔ when x = -5 25 -15 = 10 ✔
Exercises
Solve these quadratic equations by factorising
1. x2 - 8x +15 = 0
x = x =
2. x2 – 8x + 12 = 0
x = x =
3. x2 – 4x – 5 = 0
x = x =
4. x2 + 2x – 24 = 0
x = x =
5. x2 + 4x – 12 = 0
x = x =
6. x 2 - 6x – 16 = 0
x = x =
Information
If the co-efficient of the x2 does not = 1, the process of factorisation becomes harder.
Example
Solve 2x2 + 6x - 8 = 0 by factorising.
Step 1
The equation is already in the format ax2 + bx + c = 0.
Step 2
Look for 2 numbers which when multiplied together give -8 and when +/- give 6 (but in this case remember to multiply by the coefficient of x2).
-4 x 2 = -8 Or
-2 x 4 = -8
4 x 2 = 8 and 8 - 2 = 6
Step 3
In this instance there is the coefficient of the x2 term ‘a’ to consider which is 2. This makes our brackets:
(2x )(x ) = 0
Try putting -2 in the first bracket and +4 in the second bracket.
(2x - 2)(x + 4) = 0
Expand the brackets to check the signs are correct and you get back to the original equation.
(2x - 2)(x + 4) 2x2 + 8x - 2x – 8 2x2 + 6x – 8
If this does not work, try the other way round.
Step 4
Now solve the equation.
(2x – 2)(x + 4) = 0 (2x – 2) = 0
2x = 2 x = 1 or
(x + 4) = 0 x = -4
Step 5
Check your answers. 2x2 + 6x - 8 = 0 When x = 1
2 + 6 – 8 = 0 ✔ When x = -4 32 – 24 – 8 = 0✔
Exercises
Solve these quadratic equations by factorising.
1. 5x2 + 55x + 50 = 0
x = x =
2. 2x2 – 6x – 8 = 0
x = x =
3. Stanley has a vegetable garden which is 2x2 metres long and 3x metres wide.
The area is 48 m².
What is the perimeter of the vegetable garden? m
Information
Solve Quadratic Equations by completing the square
Factorising a quadratic is only a practical approach if the solutions are integer values.
Attempting to factorise a quadratic when the solutions are not integer values could prove an impossible undertaking. If you are having difficulty factorising a quadratic or if the question asks you to give answers to a specified number of decimal places or to leave the answer in surd form, you should use the following method called Completing the Square.
To solve a quadratic equation using this method, the equation is written as a perfect square plus another term.
To solve a quadratic by completing the square:
Step 1
Rearrange the equation into the form ax2 + bx = c if it is not already in this form.
Step 2
If the x2 coefficient does not equal 1.
Divide both sides of the equation by a.
Step 3
Find the number required to make the left hand side of the equation a perfect square.
This is done by diving the b term by 2 and then squaring the result.
Step 4
Add this number to both sides of the equation and write the left hand side of the equation as a square number.
Step 5
Solve this equation by taking the square root of both sides of the equation.
Example 1
Solve the equation x2 + 6x - 5 = 0 by completing the square.
Step 1
Firstly, rearrange the equation so the constant term (5) is on the right.
x2 + 6x - 5 = 0 x2 + 6x = 5
Step 2
The coefficient of x2 is 1 so we do not need to divide by anything.
Step 3
Find the number needed to make x2 + 6x into a perfect square and add this number to both sides.
To find this number, take the coefficient of x, halve it and then square the result.
The coefficient of 6x is 6 Half of 6 is 3
and 32 = 9.
So 9 is the number needed to make x2 + 6x a perfect square.
Step 4
Add 9 to both sides of the equation.
x2 + 6x + 9 = 5 + 9 x2 + 6x + 9 = 14
x2 + 6x + 9 is a perfect square.
So rewrite x2 + 6x + 9 as a square.
(x + 3)2 = 14
Step 5
Take the square root of both sides (remember +/-) and subtract 3 from both sides.
x + 3 = +/- √14 Solve for x
x = 0.742 or -6.742
Example 2
Solve 2x2 + 8x – 2 = 0.
Step 1
Rearrange to form ax2 + bx = c 2x2 + 8x – 2 = 0.
2x2 + 8x = 2
Step 2
Divide by the coefficient of x2. 2x2 + 8x = 2
x2 + 4x = 1
Step 3
Find the number required to make the left hand side a perfect square.
Coefficient of x = 4 half of 4 = 2
22 = 4
so the number required to make a perfect square is 4.
Step 4
Add 4 to both sides.
x2 + 4x + 4 = 1 + 4 simplify
(x + 2)2 = 5
Step 5
Take the square root on both sides of the equation.
x + 2 = ±√5
Subtract 2 from both sides of the equation.
x = -2 ± √5
x = -4.236 or 0.236
Exercises
1. Solve the equation x2 + 10x + 3 = 0 by completing the square.
Give your answer to 2 decimal places.
x = or x =
2. Solve the equation x2 + 4x – 1 = 0.
Give your answer to 3 decimal places.
x = or x =
3. Solve 2x2 + 4x - 2 = 0.
Give your answer to 3 decimal places.
x = or x =
4. Solve 2x2 – 12x + 6 = 0.
Give your answer to 3 decimal places.
x = or x =
Information
Solve Quadratic Equations by using the quadratic formula
Quadratic equations can be solved using the quadratic formula.
x = -b +
2a
a, b and c are the coefficients of the terms in the quadratic equation ax2 + bx + c = 0.
The quadratic formula can be used to find solutions which have whole number answers but it is easier to factorise these quadratics.
Try to factorise the equation first and, if the equation can’t be factorized, then use the formula.
If the solution is negative, check the context of the question as a negative solution may not make sense in that context.
Example
Solve the equation x2 + 3x – 2 = 0.
Give your answer to 3 decimal places.
In terms of ax2 + bx + c = 0 a = 1 b = 3 c = -2
x = -b +
2a
Substitute values for a, b and c into the formula.
Be careful with the signs of the values of a, b and c.
Remember the rules for multiplying and adding negative numbers when solving the formula.
x = -3 + 2
b2 – 4ac
b2 – 4ac
9 + 8
X = -3 + 2
X = -3.562 or 0.562
Exercises
1. Solve the following quadratic equation using the formula.
Give your answer to 3 decimal places.
2x2 -5x + 1 = 0
x = or x =
2. Solve the following quadratic equation using the formula.
Give your answer to 3 decimal places.
3x2 – 7x - 1 = 0
x = or x =
3. Solve the following quadratic equation using the formula.
Give your answer to 3 decimal places.
2x2 -4x – 5 = 0 x = or x =
17
4. Solve the equation below using any method and check the context of your answer.
A triangle has a height of x cm and a base length of (x + 3) cm.
The area of the triangle is 44 cm2
Solve this equation to find the height and the base length of the triangle.
Height = cm
Base length = cm
