Academic Skills Advice
Solving Quadratics
A quadratic equation is one where the highest power is 2.
E.g.
3𝑥 2 − 7𝑥 − 6 = 0
There are 3 ways of solving them and you can choose whichever method you prefer.
Factorising:
making 2 brackets.
Formula:
uses the quadratic formula.
Completing the square: making 1 bracket (squared).
This lesson will look at the methods of using the formula and completing the square.
(See the next lesson for how to factorise.)
Coefficients:
Before you can solve quadratics it is important that you know what the word coefficient
means as it will be used throughout this lesson.
The coefficient of 𝑥 2 is the number in front of 𝑥 2 .
The coefficient of 𝑥 is the number in front of 𝑥.
etc
E.g.
3𝑥 2 − 7𝑥 − 6 = 0
The coefficient
of 𝑥 2
The coefficient
of 𝑥
The coefficient of 𝑥 2 is 𝟑
The coefficient of 𝑥 is −𝟕
Always be careful to include the correct sign with the coefficient:
(e.g. The coefficient of 𝑥 is – 𝟕 and not 𝟕).
© H Jackson 2011 / ACADEMIC SKILLS
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Using the Formula
𝑥=
−𝑏 ± √𝑏 2 − 4𝑎𝑐
2𝑎
The above formula is used to solve a quadratic equation in the form: 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎.
Notice that to use this method the equation must always equal 0.
It is good practice to get into the habit of writing down what 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are equal to and then
substituting them into the formula.
Examples:

Use the formula to solve: 𝟐𝒙𝟐 + 𝟑𝒙 − 𝟓 = 𝟎
In the above:
𝑎=2
𝑏=3
𝑐 = −5
Substitute a, b and c into the formula (being careful to use the correct signs).
𝑥=
𝑥=
𝑥=
𝑥=
−(3)±√(3)2 −4×(2)×(−5)
2×(2)
Notice that every letter has been
replaced by a bracket with the
correct number in it.
−3±√9+40
4
−3±√49
4
−3+7
4
or
−3−7
4
5
𝑥 = 1 𝑜𝑟 𝑥 = − 2

Use the formula to solve: −𝟑𝒙𝟐 − 𝟓𝒙 − 𝟐 = 𝟎
In the above:
𝑎 = −3,
𝑥=
𝑐 = −2
−(−5)±√(−5)2 −4×(−3)×(−2)
2×(−3)
𝑥=
5±√25−24
−6
𝑥=
5±√1
−6
𝑥 = −1 or −
© H Jackson 2011 / ACADEMIC SKILLS
𝑏 = −5,
2
3
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Completing the Square
To use this method you must always make sure that the coefficient of 𝑥 2 is 1.
The method:
Make the coefficient of 𝑥 2 = 1
Half the coefficient of 𝑥
Make a bracket with (𝑥+ half the coefficient of 𝑥) and square it.
Subtract (half the coefficient of 𝑥)2
Rearrange and solve.
The method might sound complicated but try it – it’s not as difficult as it looks.
Examples:

Solve:
𝒙𝟐 − 𝟔𝒙 + 𝟕 = 𝟎 by completing the square.
Make the coefficient of 𝑥 2 = 1:
𝑥 2 − 6𝑥 + 7 = 0
(no change)
Half the coefficient of 𝑥:
−3
2
Make a bracket and square it.
(𝑥 − 3)
Subtract (half the coefficient of 𝑥)2:
(𝑥 − 3)2 − 32
Rearrange and solve:
(𝑥 − 3)2 − 32 + 7 = 0
(𝑥 − 3)2 − 9 + 7 = 0
Notice that it’s this bit that is
subtracted.
(𝑥 − 3)2 − 32 + 7 = 0
Don’t forget to
include the original 7.
(𝑥 − 3)2 − 2 = 0
(Tidy up)
(𝑥 − 3)2 = 2
(Square root each side)
𝑥 − 3 = ±√2
(Rearrange)
𝑥 = ±√2 + 3
Remember there is a
+ve and –ve answer.
𝑥 = 4.4 𝑜𝑟 𝑥 = 1.6
© H Jackson 2011 / ACADEMIC SKILLS
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
Solve:
𝟒𝒙𝟐 + 𝟏𝟔𝒙 − 𝟖 = 𝟎 by completing the square.
Make the coefficient of 𝑥 2 = 1:
𝑥 2 + 4𝑥 − 2 = 0
(÷ everything by 4)
Half the coefficient of 𝑥:
2
Notice that it’s this bit that is
subtracted.
(𝑥 + 2)2 − 22 − 2 = 0
2
Make a bracket and square it.
(𝑥 + 2)
Subtract (half the coefficient of 𝑥)2:
(𝑥 + 2)2 − 22
Rearrange and solve:
(𝑥 + 2)2 − 22 − 2 = 0
(𝑥 + 2)2 − 6 = 0
Don’t forget to include
the original -2.
(𝑥 + 2)2 = 6
𝑥 + 2 = ±√6
𝑥 = ±√6 − 2
𝑥 = 0.45 𝑜𝑟 𝑥 = −4.45

Solve:
𝟑𝒙𝟐 + 𝟕𝒙 − 𝟔 = 𝟎 by completing the square.
Make the coefficient of 𝑥 2 = 1:
7
𝑥2 + 3 𝑥 − 2 = 0
(÷ everything by 3)
Half the coefficient of 𝑥:
(don’t let the fraction put you off,
just carry on as normal)
7
6
7 2
Make a bracket and square it.
(𝑥 + 6)
Subtract (half the coefficient of 𝑥)2:
(𝑥 + 6) − (6) − 2
Rearrange and solve:
(𝑥 + ) −
7 2
7 2
7 2
121
6
36
7 2
121
(𝑥 + 6) =
7
=0
36
121
𝑥 + 6 = ±√ 36
121
7
𝑥 = ±√ 36 − 6
𝑥 = 0.67 𝑜𝑟 𝑥 = −3
© H Jackson 2011 / ACADEMIC SKILLS
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