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In this worksheet, students will learn how to solve quadratic equations with three terms by rearranging and finding square roots as whole numbers and decimals (no factorising required).

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:

### 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?'

You may already be familiar with this type of puzzle and how it can be expressed algebraically.

These sorts of problems are best solved by writing them as equations.

If we use x (or any letter) to stand for original number, we can write the puzzle without using any words but using numbers, letters and symbols as: x² + 9 = 90

Remember, that when we multiply a number by itself, we are squaring it.

Any equation which contains an as the highest power of x (so we have no x³ for example) is called a quadratic equation.

Solving this special type of equation is a very important area of maths.

Now, to find out what x is, we need to solve the equation.

This means getting it into the form: x = ...

Remember that whatever we do to one side of the equation, we must do to the other.

So, firstly we need to remove the + 9 by doing the inverse to both sides which is - 9.

This leaves us with:

x² = 81

Now, we need to remove the 'squaring' by doing the inverse of this 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 both 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 can set out our working like this:

Let's try another example now.

Here is another equation: 2x² + 2 = 100

In order to solve this equation, first, we must subtract 2 from both sides:

2x² = 98

Now we need to be careful about the order we do the next two inverses.

2x² means 2 times , so we need to apply the inverse of × 2 next which is ÷ 2:

x² = 49

Finally, we square root both sides to give:

x = ±7

We set out our working like this:

If our last step here does not have a whole number square root, we will need to use a calculator.

There are some questions like this in the activity, so you should grab a calculator before you start.

In this activity, we will solve quadratic equations with three terms by rearranging and calculating square roots of whole numbers and decimals.

Solve the equation below:

x² - 7 = 57

x = ±6

x = ±7

x = ±8

x = ±9

Solve the equation below:

n² + 12 = 21

n = ±1

n = ±2

n = ±3

n = ±4

Match each equation below with its correct solution.

## Column B

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

Match each equation below with its correct solution.

## Column B

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

At least one of the equations below has a solution of:

x = ±1

Tick all the equations which fit with this solution.

2x² + 8 = 9

3x² - 1 = 2

4x² + 3 = 11

5x² - 6 = 1

6x² - 8 = -2

Solve the equation:

10y² - 94 = 1116

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 below:

2x² - 4 = 10

Choose the solutions from the list below, all given to 1 decimal place.

1.7

2.6

-2.6

-1.7

You'll need a calculator for this question.

Match each equation below to its correct positive solution to one decimal place.

## Column B

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

3n² -18 = -5

Which of the options below is the correct negative solution to this equation to one decimal place?

-2.0

-2.1

-2.7

-2.8

Solve the equation below:

7x² - 19 = 64

Write both solutions in the spaces below, giving your answers to 2 decimal places.

-2.0

-2.1

-2.7

-2.8

• Question 1

Solve the equation below:

x² - 7 = 57

x = ±8
EDDIE SAYS
Remember that whatever we do to one side of the equation, we must do exactly the same to the other side. We want to get rid of the '- 7' on the left-hand side, so we need to do the inverse (or opposite) of this to both sides. The opposite of - 7 is + 7. This leaves us with: x² = 64 Then we square root, both sides giving us: x = ±7 How did you get on with this first challenge? Review the examples in the Introduction before you move on to the rest of the activity.
• Question 2

Solve the equation below:

n² + 12 = 21

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

Match each equation below with its correct solution.

## Column B

p² + 48 = 73
p = ±5
p² - 27 = 73
p = ±10
p² + 37 = 73
p = ±6
EDDIE SAYS
Work through each equation one at a time, rearranging each of the form: x = ... As there are three equations to solve here we will shorten our working out: p² + 48 = 73 → - 48 then square root → p = ±5 p² - 27 = 73 → + 27 then square root → p = ±10 p² + 37 = 73 → - 37 then square root → p = ±6 Did you match these all accurately?
• Question 4

Match each equation below with its correct solution.

## Column B

5d² = 80
d = ±4
3d² - 67 = 80
d = ±7
4d² + 74 = 80
d = ±2
EDDIE SAYS
We'll just show our 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 spot the pairs here?
• Question 5

At least one of the equations below has a solution of:

x = ±1

Tick all the equations which fit with this solution.

3x² - 1 = 2
6x² - 8 = -2
EDDIE SAYS
We can check each equation by substituting in 1 (or -1) in the place of x to see if it gives the correct answer. 2x² + 8 = 9 → 2 × 1² + 8 = 2 + 8 = 10 (not a match) 3x² - 1 = 2 → 3 × 1² - 1 = 3 - 1 = 2 (matches) 4x² + 3 = 22 → 4 × 1² + 3 = 4 + 3 = 7 (not a match) 5x² - 6 = 1 → 5 × 1² - 6 = 5 - 6 = -1 (not a match) 6x² - 8 = -2 → 6 × 1² - 8 = 6 - 8 = -2 (matches) So two of these equations work with the solution x = ±1. Were you able to find them both?
• Question 6

Solve the equation:

10y² - 94 = 1116

EDDIE SAYS
There are some big numbers here, but our method is the same. 10y² - 94 = 1116 10y² - 94 + 94 = 1116 + 94 10y² = 1210 10y² ÷ 10 = 1210 ÷ 10 y² = 121 √y² = √121 y = ±11 Another way to think about this answer is that y could be +11 or -11. Did you type those options into the blanks accurately?
• Question 7

You will need a calculator for this question.

Solve the equation below:

2x² - 4 = 10

Choose the solutions from the list below, all given to 1 decimal place.

2.6
-2.6
EDDIE SAYS
The working for this equation is as follows: 2x² - 4 = 10 2x² - 4 + 4 = 10 + 4 2x² = 14 2x² ÷ 2 = 14 ÷ 2 x² = 7 √x² = √7 Type √7 into your calculator. You will reach a long answer - what is this decimal rounded to one decimal place? In this list, +2.6 or -2.6 are both possible answers.
• Question 8

You'll need a calculator for this question.

Match each equation below to its correct positive solution to one decimal place.

## 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? Are these getting more challenging?
• Question 9

3n² -18 = -5

Which of the options below is the correct negative solution to this equation to one decimal place?

-2.1
EDDIE SAYS
Be careful with the positive and negative symbols here! 3n² -18 = -5 3n² -18 + 18 = -5 + 18 3n² = 13 3n² ÷ 3 = 13 ÷ 3 n² = 4.33.... (Don't round this off yet!) Let's find the square root of this long decimal number now: n = ±2.1 (1 d.p) So the correct negative solution is: -2.1
• Question 10

Solve the equation below:

7x² - 19 = 64

Write both solutions in the spaces below, giving your answers to 2 decimal places.

EDDIE SAYS
This is the last and most difficult challenge! Here's our working: 7x² - 19 = 64 7x² - 19 + 19 = 64 + 19 7x² = 83 7x² ÷ 7 = 83 ÷ 7 x² = 11.857..... (Don't round this off yet!) √x² = √11.857..... x = ±3.44 (2 d.p.) Well, that's it. How did you do? Why not try another quadratic equation activity now to really perfect your skills?
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