KS3 Maths · Key Stage 3 · Unit 9: Quadratics and Simultaneous Equations

Lines through two points, double brackets, algebraic fractions, quadratic graphs and simultaneous equations

Year 9About 85 minutesNo calculator89 marks

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NameClassDate

Before you start

Recall

You will need these in every question below.

A1
Expand $3 (2 x - 5)$ .
[1 mark]
Answer

A2
Factorise $8 x + 12$ fully.
[1 mark]
Answer
A3
Solve $5 x - 7 = 2 x + 8$ .
[2 marks]
Answer
A4
Write down the gradient and the $y$ -intercept of the line $y = 2 x - 9$ .
[2 marks]
Answergradient $\dots\dots$ $y$ -intercept $\dots\dots$
A5
Work out the value of $x^{2} - 3 x$ when $x = - 2$ .
[1 mark]
Answer

Examples, then your turn

Core

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

Example 1

A straight line passes through the points $(1, 5)$ and $(3, 9)$ . Find the equation of the line.

Hint The gradient is the change in

y

divided by the change in

x

. Then put one of the points into

y = m x + c

to find

c

B1
A straight line passes through the points $(1, 7)$ and $(4, - 2)$ . Find the equation of the line.
[3 marks]
Answer

Example 2

Expand and simplify $(x + 8) (x - 3)$ .

Hint Multiply each term in the first bracket by each term in the second. There are four products. Then collect the two

x

terms.

B2
Expand and simplify $(x - 3) (x + 9)$ .
[2 marks]
Answer

Example 3

Factorise $x^{2} + 5 x - 14$ .

Hint Find two numbers that multiply to

- 14

and add to

5

B3
Factorise $x^{2} - 4 x - 45$ .
[2 marks]
Answer

Example 4

Simplify $\frac{x ^{2} + 8 x + 15}{x ^{2} - 25}$ .

Hint Factorise the top and the bottom first. You can only cancel a whole bracket that is a factor of both.

B4
Simplify $\frac{x ^{2} - 5 x - 6}{x ^{2} - 36}$ .
[3 marks]
Answer

Practice

Core

Each question asks for something different. Show your working.

C1
Factorise $x^{2} - 81$ .
[1 mark]
Answer

C2
The graph of $y = (x - 3) (x + 5)$ is a curve.
(a)
Write down the two values of $x$ where the curve crosses the $x$ -axis.
[2 marks]
Answer
(b)
Work out where the curve crosses the $y$ -axis.
[1 mark]
Answer
(c)
Work out the coordinates of the turning point of the curve.
[2 marks]
Answer
Total for C2: 5 marks
C3
Simplify $\frac{x ^{2} + 4 x - 32}{x - 4}$ .
[2 marks]
Answer
C4
Write $\frac{3}{x + 1} - \frac{2}{x + 4}$ as a single fraction. Simplify the top.
[3 marks]
Answer
C5
The lines $y = 4 x - 3$ and $y = 2 x + 7$ cross at one point. Work out the coordinates of that point.
[3 marks]
Answer
C6
How many solutions do the simultaneous equations $y = 2 x + 3$ and $4 y = 8 x - 5$ have? Give a reason for your answer.
[2 marks]
Answer $\dots\dots$ solutions, because $\dots\dots$
C7
A straight line passes through $(1, 1)$ and $(4, 10)$ . The point $(6, k)$ is also on the line. Work out the value of $k$ .
[2 marks]
Answer
C8
The graph of $y = (x + 2) (x - 1) (x - 5)$ is a cubic curve.
(a)
Write down the values of $x$ where the curve crosses the $x$ -axis.
[2 marks]
Answer
(b)
Work out where the curve crosses the $y$ -axis.
[1 mark]
Answer
Total for C8: 3 marks
C9
Expand and simplify $(x + 4)^{2} - (x - 2) (x + 3)$ .
[3 marks]
Answer

Test-style questions

Stretch

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Mia says, " $(x + 5)^{2}$ is the same as $x^{2} + 25$ ."
Mia is wrong. Show that she is wrong, and write down the correct expansion of $(x + 5)^{2}$ .
[2 marks]
Answercorrect expansion $\dots\dots$

D2
A rectangle is $(x + 7)$ cm long and $(x - 2)$ cm wide. Its area is $52$ cm².
Not drawn accurately
(a)
Show that $x^{2} + 5 x - 66 = 0$ .
[2 marks]
(b)
Solve $x^{2} + 5 x - 66 = 0$ to find the value of $x$ . You must show your working.
[3 marks]
Answer
(c)
Work out the perimeter of the rectangle.
[1 mark]
Answer
Total for D2: 6 marks
D3
The points $A (- 2, 5)$ and $B (4, - 7)$ are shown on the grid.
(a)
Find the equation of the straight line through $A$ and $B$ .
[3 marks]
Answer
(b)
Show that the point $C (10, - 19)$ is on the same line.
[2 marks]
Total for D3: 5 marks
D4
Gym A charges a joining fee of £20 and then £6 for each visit. Gym B has no joining fee and charges £8 for each visit.
(a)
For how many visits do the two gyms cost the same? What is that cost? You must show your working.
[3 marks]
Answer $\dots\dots$ visits £ $\dots\dots$
(b)
Sana plans to make 15 visits. Which gym is cheaper for her, and by how much?
[2 marks]
AnswerGym $\dots\dots$ by £ $\dots\dots$
Total for D4: 5 marks
D5
The curve $y = x^{2} + b x + c$ crosses the $x$ -axis at $x = - 2$ and $x = 8$ .
(a)
Work out the values of $b$ and $c$ .
[2 marks]
Answer $b = \dots\dots$ $c = \dots\dots$
(b)
Work out the coordinates of the turning point of the curve.
[2 marks]
Answer
Total for D5: 4 marks
D6
Simplify fully $\frac{x ^{2} + 6 x + 9}{x - 1} \div \frac{x + 3}{x ^{2} - 1}$ .
[4 marks]
Answer
D7
The line $y = x + 7$ and the curve $y = (x + 1)^{2}$ cross at two points. Work out the coordinates of both points. You must show your working.
[4 marks]
Answer
D8
Use the difference of two squares to work out $8 5^{2} - 1 5^{2}$ without a calculator. You must show your working.
[2 marks]
Answer
D9
The curve $y = (x - a) (x + 2) (x - 4)$ crosses the $y$ -axis at $(0, 24)$ . Work out the value of $a$ .
[2 marks]
Answer

Challenge

UKMT level

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
The expression $x^{2} + b x + c$ factorises into two brackets $(x + p) (x + q)$ , where $p$ and $q$ are positive whole numbers. Also $b + c = 20$ .
Work out the values of $b$ and $c$ .
[4 marks]
Hint 1 · What to try
Expand $(x + p) (x + q)$ . What are $b$ and $c$ in terms of $p$ and $q$ ?
Hint 2 · The first line
$b = p + q$ and $c = pq$ , so $p + q + pq = 20$ . Add 1 to both sides.
Hint 3 · The full method
$pq + p + q + 1 = (p + 1) (q + 1) = 21$ . The only way with $p$ and $q$ positive is $3 \times 7$ , so $p = 2$ and $q = 6$ .
Answer $b = \dots\dots$ $c = \dots\dots$

E2
The line $y = m x + 12$ has a negative gradient $m$ . The line, the $x$ -axis and the $y$ -axis make a triangle with an area of 36 square units.
Work out the value of $m$ .
[3 marks]
Hint 1 · What to try
The line crosses the $y$ -axis at 12. Where does it cross the $x$ -axis?
Hint 2 · The first line
It crosses the $x$ -axis where $m x + 12 = 0$ . The triangle has height 12, so its base must be 6.
Hint 3 · The full method
The line meets the $x$ -axis at $(6, 0)$ , so $0 = 6 m + 12$ and $m = - 2$ .
Answer
E3
The curve $y = x^{2} - 6 x + k$ touches the $x$ -axis at exactly one point.
Work out the value of $k$ .
[3 marks]
Hint 1 · What to try
If the curve touches the $x$ -axis once, the turning point is on the $x$ -axis.
Hint 2 · The first line
The curve is symmetrical about the line $x = 3$ , so the turning point is where $x = 3$ .
Hint 3 · The full method
At $x = 3$ , $y = 9 - 18 + k$ . This must be 0, so $k = 9$ .
Answer
E4
The lines $y = 2 x + 3$ and $y = 15 - x$ and the $y$ -axis make a triangle.
Work out the area of the triangle.
[4 marks]
Hint 1 · What to try
Find the three corners. Two of them are on the $y$ -axis.
Hint 2 · The first line
The lines meet the $y$ -axis at $(0, 3)$ and $(0, 15)$ . They meet each other where $2 x + 3 = 15 - x$ .
Hint 3 · The full method
They meet at $(4, 11)$ . Take the side on the $y$ -axis, of length 12, as the base. The height is 4, so the area is $\frac{1}{2} \times 12 \times 4$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the equation of a straight line from two points on it, and check whether a third point is on the same line.
I can expand two brackets, factorise a quadratic expression, and use the difference of two squares.
I can simplify an algebraic fraction by factorising, and add, subtract, multiply and divide algebraic fractions.
I can find the roots, the $y$ -intercept and the turning point of a quadratic graph, and the roots of a cubic graph.
I can solve two simultaneous equations by finding where their graphs meet, and say when there is no solution.

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