KS3 Maths · Year 8 · Chapter 11: Solving equations graphically · Lesson 3

Solving quadratic equations by drawing graphs

Year 8About 70 minutesNo calculator45 marks

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Quick questions to start.

A1
Last lesson
The lines $y = 2 x + 1$ and $y = 7$ cross at one point. Write down the coordinates of that point.
[1 mark]
Answer
A2
Needed today
Work out the value of $x^{2} - 3 x$ when $x = - 2$ .
[1 mark]
Answer
A3
From chapter 9
A length is $7.4$ cm, correct to $1$ decimal place. Write down its lower bound.
[1 mark]
Answer
A4
Year 7
Expand $x (x + 5)$ .
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

Complete the table of values for $y = x^{2} + x - 2$ . Use it to solve $x^{2} + x - 2 = 0$ .

Hint For each

x

, add

x^{2}

and

x

, then take away

2

. The solutions are where

y = 0

B1
Complete the table of values for $y = x^{2} - 3 x + 2$ . Use it to solve $x^{2} - 3 x + 2 = 0$ .
[3 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Example 2

The graph of $y = x^{2} - 2 x - 3$ is drawn. (a) Use it to solve $x^{2} - 2 x - 3 = 5$ . (b) How many solutions do $x^{2} - 2 x - 3 = - 4$ and $x^{2} - 2 x - 3 = - 5$ have?

Hint Draw each line

y = 5

y = - 4

y = - 5

. Count where each meets the curve. The lowest point is

(1, - 4)

B2
The graph of $y = x^{2} - 6 x + 5$ is drawn. (a) Use it to solve $x^{2} - 6 x + 5 = - 3$ . (b) How many solutions does $x^{2} - 6 x + 5 = - 4$ have?
[3 marks]
Answer

Example 3

The graph of $y = x^{2} - x - 1$ is drawn. Use it to solve $x^{2} - x - 3 = 0$ . Give your answers to $1$ decimal place.

Hint Add

2

to both sides to get

x^{2} - x - 1 = 2

. Draw the line

y = 2

. Each crossing is between gridlines: estimate how far along the square it is.

B3
The graph of $y = x^{2} - 2 x - 2$ is drawn. Draw a suitable line and use it to solve $x^{2} - 2 x - 7 = 0$ . Give your answers to $1$ decimal place.
[3 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Practice

Year 8

C1
The graph of $y = x^{2} - 6 x + 7$ is drawn. Write down the coordinates of its turning point, and the equation of its line of symmetry.
[2 marks]
Answer
C2
One solution of $x^{2} - 8 x + 12 = 0$ is $x = 2$ . The line of symmetry of $y = x^{2} - 8 x + 12$ is $x = 4$ . Work out the other solution.
[2 marks]
Not drawn accurately
Answer

C3
The lowest point of $y = x^{2} + 4 x + 1$ is $(- 2, - 3)$ . For which value of $k$ does $x^{2} + 4 x + 1 = k$ have exactly one solution? For which values of $k$ does it have no solution?
[2 marks]
Not drawn accurately
Answer
C4
The line $y = 3$ and the curve $y = x^{2} - 5 x + 7$ are drawn. Which equation do their crossing points solve? Write it in the form $x^{2} + b x + c = 0$ , and write down its solutions.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Jo uses the graph of $y = x^{2} - x - 6$ to solve $x^{2} - x - 6 = 0$ . She says, "The solution is $x = - 6$ , because that is where the curve crosses the $y$ -axis."
Explain what Jo has done wrong. Find the correct solutions.
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

D2
A rectangle is $x$ cm wide. Its length is $1$ cm more than its width. Its area is $5$ cm².
(a)
Show that $x^{2} + x = 5$ .
[1 mark]
(b)
Work out the value of $x^{2} + x$ when $x = 0$ , $1$ and $2$ . Plot the points on the grid and draw the graph of $y = x^{2} + x$ through them.
[2 marks]
(c)
Use your graph to find the width of the rectangle. Give your answer to $1$ decimal place.
[2 marks]
Answer
Total for D2: 5 marks
D3
The graph of $y = x^{2} + 2 x + 1$ is drawn on the grid.
(a)
Use the graph to solve $x^{2} + 2 x + 1 = 9$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
(b)
By drawing a suitable line, solve $x^{2} + 2 x - 2 = 0$ . Give your answers to $1$ decimal place.
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
(c)
Explain why $x^{2} + 2 x + 1 = - 2$ has no solution.
[2 marks]
Total for D3: 6 marks
D4
A ball is thrown up. Its height, $h$ metres, after $t$ seconds is $h = 4 t - t^{2}$ . The graph of $h$ against $t$ is drawn.
(a)
At what two times is the ball $3$ m high?
[2 marks]
Answer
(b)
For how many seconds is the ball higher than $3$ m?
[1 mark]
Answer
(c)
Write down the greatest height of the ball.
[1 mark]
Answer
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The equation $x^{2} - 6 x + k = 0$ has exactly one solution. Work out the value of $k$ , and the solution.
[5 marks]
Hint 1 · What to try
Write it as $x^{2} - 6 x = - k$ . Where does the curve $y = x^{2} - 6 x$ cross the $x$ -axis?
Hint 2 · The first line
It crosses at $x = 0$ and $x = 6$ , so its line of symmetry is $x = 3$ . Work out $y$ when $x = 3$ .
Hint 3 · The full method
The lowest point is $(3, - 9)$ . One solution means the line $y = - k$ just touches it.
Answer $k = \dots\dots$ $x = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can complete a table of values and read solutions where $y = 0$ .
I can draw a line $y = k$ on a curve and read the solutions.
I can say whether a quadratic equation has two, one or no solutions.

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