A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 2: Quadratics · Lesson 3

Quadratic graphs and the discriminant

70 marks

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Quick questions to start.

A1
Needed today
Write $x^{2} - 8 x + 23$ in the form $(x + p)^{2} + q$ .
[2 marks]
Answer
A2
Needed today
Solve $3 x^{2} - 5 x - 2 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A3
Last lesson
Find the four roots of $f (x) = x^{4} - 10 x^{2} + 9$ .
[2 marks]
Answer
A4
From chapter 1
Rationalise the denominator of $\frac{6}{3 - 3}$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Sketch $y = x^{2} + 2 x - 35$ . Show the axis crossings and the turning point.

Hint Factorise. The turning point is half-way between the roots.

B1
Sketch $y = 2 x^{2} - x - 6$ , showing where it meets the axes and its turning point.
[3 marks]

Example 2

Sketch the graph of $y = 4 x - x^{2} - 7$ . Give the coordinates of the turning point and show where the graph meets the axes.

Hint Write it as

- x^{2} + 4 x - 7

and take out a factor of

- 1

before completing the square. Check the discriminant to see if it meets the

x

-axis.

B2
Write $6 x - 3 x^{2} - 5$ in the form $a (x + p)^{2} + q$ . Hence sketch $y = 6 x - 3 x^{2} - 5$ , showing its turning point and where it meets the axes.
[4 marks]

Example 3

The graph of $y = a x^{2} + b x + c$ meets the $x$ -axis at $(- 1, 0)$ and $(4, 0)$ and the $y$ -axis at $(0, - 8)$ . Find $a$ , $b$ and $c$ .

Hint If the roots are

- 1

and

4

, then

y = a (x + 1) (x - 4)

. Use the

y

-intercept to find

a

B3
The graph of $y = a x^{2} + b x + c$ has a maximum point at $(- 2, 7)$ and passes through $(0, - 5)$ . Find $a$ , $b$ and $c$ .
[3 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$

Example 4

Find the values of $k$ for which $(k - 1) x^{2} + 6 x + (k + 7) = 0$ has equal roots.

Hint Equal roots means

b^{2} - 4 a c = 0

. Here

a = k - 1

b = 6

and

c = k + 7

. You get a quadratic in

k

B4
Find the values of $k$ for which $x^{2} + (k + 3) x + 4 k = 0$ has equal roots.
[3 marks]
Answer $k = \dots\dots$ or $k = \dots\dots$

Example 5

Find the range of values of $k$ for which $3 x^{2} - 6 x + k = 0$ has two distinct real roots.

Hint Two distinct real roots means

b^{2} - 4 a c > 0

. The answer is an inequality.

B5
Find the range of values of $k$ for which $x^{2} - 4 x + (2 k - 1) = 0$ has no real roots.
[3 marks]
Answer $k \dots\dots$

Practice

A-Level

C1
$f (x) = 7 + 3 x - 2 x^{2}$ . Find the value of the discriminant of $f (x)$ , and hence state how many times the graph of $y = f (x)$ meets the $x$ -axis.
[2 marks]
Answer

C2
The sketch shows the graph of $y = a x^{2} + b x + c$ . State, with a reason in each case, whether each of $a$ , $b$ , $c$ and $b^{2} - 4 a c$ is positive, negative or zero.
[4 marks]
C3
Find the range of values of $k$ for which the line $y = k$ does not meet the curve $y = x^{2} - 10 x + 33$ .
[3 marks]
Answer $k \dots\dots$

Exam-style questions

Edexcel exam style

D1
Leah writes: $3 x^{2} - 18 x + 22 = 3 (x - 3)^{2} + 13$ , so the minimum point of $y = 3 x^{2} - 18 x + 22$ is $(3, 13)$ .
$f (x) = 3 x^{2} - 18 x + 22$ . Leah is wrong.
(a)
Describe the mistake Leah has made.
[2 marks]
(b)
Write $f (x)$ in the form $a (x + p)^{2} + q$ , where $a$ , $p$ and $q$ are integers. Hence write down the coordinates of the turning point of $y = f (x)$ and the equation of its line of symmetry.
[5 marks]
Answer $f (x) = \dots\dots$ $(\dots\dots, \dots\dots)$ $x = \dots\dots$
(c)
Hence solve $f (x) = 0$ , giving your answers in the form $\frac{a \pm b}{c}$ , where $a$ , $b$ and $c$ are integers.
[2 marks]
Answer $x = \dots\dots$
Total for D1: 9 marks

D3
$h (x) = 3 x^{2} + (k - 1) x - (k + 2)$ , where $k$ is a real constant.
(a)
Find the discriminant of $h (x)$ in terms of $k$ , simplifying your answer.
[3 marks]
Answer
(b)
Hence prove that $h (x)$ has real roots for every value of $k$ .
[2 marks]
(c)
Find the value of $k$ for which $h (x)$ has equal roots, and find this repeated root.
[3 marks]
Answer $k = \dots\dots$ $x = \dots\dots$
Total for D3: 8 marks
D4
The sketch shows $y = a x^{2} + b x + c$ , with maximum point $(2, 9)$ and $y$ -intercept $(0, 1)$ .
(a)
Find the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $c = \dots\dots$
(b)
Hence find the exact $x$ -coordinates of the points where the graph meets the $x$ -axis. Give your answers in the form $p \pm q 2$ , where $p$ and $q$ are rational.
[3 marks]
Answer $x = \dots\dots$
Total for D4: 7 marks
D5
Given that $y = 2^{x}$ ,
(a)
show that the equation $4^{x} - k \times 2^{x + 1} + 16 = 0$ can be written as $y^{2} - 2 k y + 16 = 0$ , and hence find the values of $k$ for which this equation in $y$ has equal roots.
[4 marks]
Answer $k = \dots\dots$
(b)
For one value of $k$ the equation in $x$ has one solution. Find it, and explain why the other value of $k$ gives none.
[3 marks]
Answer $x = \dots\dots$
Total for D5: 7 marks

Extension

Stretch

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

E1
Find every integer $k$ for which the equation $x^{2} + k x + k + 8 = 0$ has two distinct roots that are both integers.
[6 marks]
Hint 1 · What to try
Call the roots $r$ and $s$ , so $x^{2} + k x + k + 8 = (x - r) (x - s)$ .
Hint 2 · The first line
$r + s = - k$ and $r s = k + 8$ . Add these: $r s + r + s = 8$ , so $(r + 1) (s + 1) = 9$ .
Hint 3 · The full method
Use the pairs $1 \times 9$ and $(- 1) \times (- 9)$ . The pairs $3 \times 3$ and $(- 3) \times (- 3)$ give equal roots.
Answer $k = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can sketch a quadratic and its turning point.
I can find a quadratic from its sketch.
I can count real roots with the discriminant.
I can find $k$ using the discriminant.

Answers and mark scheme

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