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Solve Quadratic Equations

In this worksheet, students will learn how to solve simple quadratic equations by rearranging and square rooting (no factorising required).

'Solve Quadratic Equations' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Pearson Edexcel, OCR, Eduqas, AQA

Curriculum topic:   Algebra

Curriculum subtopic:   Solving Equations and Inequalities, Algebraic Equations

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

'I think of a number, multiply it by itself then add 9 and the answer is 90. What number was I thinking of?'

If you have completed any of the Level 1 activities on linear equations you will be familiar with this type of puzzle. But if you haven't, do not worry, all will be explained. These sorts of problems are best solved by writing them as equations. If I use x (or any letter) to stand for the number I thought of, I can write the puzzle without using any words but using numbers, letters and symbols as:

x² + 9 = 90 

Remember, when we multiply a number by itself we are squaring. Any equation which contains an as the highest power of x (no x³ for example) is called a quadratic equation, and solving them is a very important area of maths. Now, to find out what number x is I solve the equation. That means getting it in the form x = ......, by doing the same to both sides. So, firstly we need to remove the +9 by doing the inverse to both sides which is -9. This gives us 

x² = 81

Now, I need to remove the 'squaring' by doing the inverse to both sides which is 'square rooting'. This gives us

x = 9

Now strictly speaking there are two square roots of 81. This is because when you square a negative number you get a positive, so 9² = 81 and (-9)² = 81. So the square root of 81 could be 9 or -9. Usually, we are just interested in the positive square root, but it is always safest to include both. Remember, the symbol for square root is √ so we write

√81 = 9 or -9      or      √81 = ±9    (the ± symbol means the answer can be positive or negative)

We set the working out to this as follows:

Here is another equation:

2x² + 2 = 100

In order to solve this equation, first, we must subtract 2 from both sides. This will give us 2x² = 98. Now we need to be careful about the order we do the next two inverses. 2x² means 2 times x² so we need to apply the inverse of x2 next which is ÷2. This gives us x² = 49. Finally, we square root both sides to give x = ±7. We set out the working as follows.

If the last step does not have a whole number square root we will need to use a calculator. There are some questions like this in the exercise. Ready to try some now? Here we go!

 

Solve the equation

x² - 7 = 57

Choose your answer from the following.

x = ±6

x = ±7

x = ±8

x = ±9

Solve the equation

n² + 12 = 21

Choose from the following solutions.

 

n = ±1

n = ±2

n = ±3

n = ±4

Match the following equations with their solutions.

Column A

Column B

p² + 48 = 73
p = ±5
p² -27 = 73
p = ±6
p² + 37 = 73
p = ±10

Which of the following equations has the solution x = ±1? You can choose more than one.

 

 

Column A

Column B

5d² = 80
d = ±4
3d² - 67 = 80
d = ±2
4d² + 74 = 80
d = ±7

Which of the following equations has the solution x = ±1? You may choose more than one.

 

2x² + 8 = 9

3x² - 1 = 2

4x² + 3 = 11

5x² - 6 = 1

6x² - 8 = -2

Solve the equation

10y² - 94 = 1116

Fill your answer in the space below.

2x² + 8 = 9

3x² - 1 = 2

4x² + 3 = 11

5x² - 6 = 1

6x² - 8 = -2

You will need a calculator for this question.

Solve the equation 

2x² - 4 = 10

Choose the solutions from the list below, all given correct to 1 decimal place. You may choose more than one.

1.7

2.6

-2.6

-1.7

You'll need a calculator for this question.

Match the equations to the positive solutions given correct to one decimal place.

 

 

 

Column A

Column B

x² - 9 = 20
x = 5.5
3x² = 94
x = 5.6
2x² + 40 = 100
x = 5.4

For the equation

3n² -18 = -5

which of the choices below is the correct negative solution? (correct to 1 decimal place.)

 

-2.0

-2.1

-2.7

-2.8

Solve the equation below, writing both solutions in the spaces below. Give your solutions correct to 2 decimal places.

7x² - 19 = 64

 

 

-2.0

-2.1

-2.7

-2.8

  • Question 1

Solve the equation

x² - 7 = 57

Choose your answer from the following.

CORRECT ANSWER
x = ±8
EDDIE SAYS
First, we add 7 to both sides. This gives us x² = 64. Then we square root both sides giving us x = ±7. Did you get it right?
  • Question 2

Solve the equation

n² + 12 = 21

Choose from the following solutions.

 

CORRECT ANSWER
n = ±3
EDDIE SAYS
Here is the working for this one: n² + 12 = 21 -12 -12 n ² = 9 √ √ n = ±3
  • Question 3

Match the following equations with their solutions.

CORRECT ANSWER

Column A

Column B

p² + 48 = 73
p = ±5
p² -27 = 73
p = ±10
p² + 37 = 73
p = ±6
EDDIE SAYS
As there are three equations to solve here we will shorten the working out. p² + 48 = 73 → -48 then square root → p = plusmn5 p² - 27 = 73 → +27 then square root → p = plusmn10 p² + 37 = 73 → -37 then square root → p = plusmn6 How did you do? Did you match them all?
  • Question 4

Which of the following equations has the solution x = ±1? You can choose more than one.

 

 

CORRECT ANSWER

Column A

Column B

5d² = 80
d = ±4
3d² - 67 = 80
d = ±7
4d² + 74 = 80
d = ±2
EDDIE SAYS
We'll just show the shortened working for each one again. 5d² = 80 → ÷5 then square root → d = ±4 3d² - 67 = 80 → +67 then ÷3 then square root → d = ±7 4d² + 74 = 80 → -74 then ÷4 then square root → d = ±2 Did you get all these correct?
  • Question 5

Which of the following equations has the solution x = ±1? You may choose more than one.

 

CORRECT ANSWER
3x² - 1 = 2
6x² - 8 = -2
EDDIE SAYS
We can check each equation by substituting in 1 (or -1) for x to see if it gives the correct answer. 2x² + 8 = 9 → 2 x 1² + 8 = 2 + 8 = 10 (wrong) 3x² - 1 = 2 → 3 x 1² - 1 = 3 - 1 = 2 (correct) 4x² + 3 = 22 → 4 x 1² + 3 = 4 + 3 = 7 (wrong) 5x² - 6 = 1 → 5 x 1² - 6 = 5 - 6 = -1 (wrong) 6x² - 8 = -2 → 6 x 1² - 8 = 6 - 8 = -2 (correct) Did you get them both?
  • Question 6

Solve the equation

10y² - 94 = 1116

Fill your answer in the space below.

CORRECT ANSWER
EDDIE SAYS
There are some big numbers here, but the method is the same. 10y² - 94 = 1116 +94 +94 10y² = 1210 ÷10 ÷10 y² = 121 √ √ y = ±11
  • Question 7

You will need a calculator for this question.

Solve the equation 

2x² - 4 = 10

Choose the solutions from the list below, all given correct to 1 decimal place. You may choose more than one.

CORRECT ANSWER
2.6
-2.6
EDDIE SAYS
The working for this equation is as follows: 2x² - 4 = 10 +4 +4 2x² = 14 ÷2 ÷2 x² = 7 √ √ x = ±2.6
  • Question 8

You'll need a calculator for this question.

Match the equations to the positive solutions given correct to one decimal place.

 

 

 

CORRECT ANSWER

Column A

Column B

x² - 9 = 20
x = 5.4
3x² = 94
x = 5.6
2x² + 40 = 100
x = 5.5
EDDIE SAYS
Here's the working for each equation. x² - 9 = 20 → +9 then square root → x = 5.4 (or -5.4) 3x² = 94 → ÷3 then square root → x = 5.6 (or -5.6) 2x² + 40 = 100 → -40 then ÷2 then square root → x = 5.5 (or -5.5) How was that one? Getting harder?
  • Question 9

For the equation

3n² -18 = -5

which of the choices below is the correct negative solution? (correct to 1 decimal place.)

 

CORRECT ANSWER
-2.1
EDDIE SAYS
Watch the negatives in this one 3n² -18 = -5 +18 +18 3n² = 13 ÷3 ÷3 n² = 4.33.... (don't round this off yet) n = ±2.1 (1 d.p) so the negative solution is -2.1
  • Question 10

Solve the equation below, writing both solutions in the spaces below. Give your solutions correct to 2 decimal places.

7x² - 19 = 64

 

 

CORRECT ANSWER
EDDIE SAYS
This is the last and most difficult. Here's the working: 7x² - 19 = 64 + 19 +19 7x² = 83 ÷7 ÷7 x² = 11.857..... (don't round this off yet) √ √ x = ±3.44 (2d.p) Well, that's it. How did you do?
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