# Quadratic equations functions

The perimeter of this rectangular field is 220 m. One side is x m as shown.

Wm

xm

(a)

Express the width (W) in terms of x.

(b)

Write an expression, in terms of x only, for the area of the field.

(c)

If the length (x) is 70 m, find the area.

Working:

Answers:

(a) …………………………………………..

(b) …………………………………………..

(c) …………………………………………..

(Total 4 marks)

1

2.

The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and has a vertex at B.

y

A

x

O

B

(a)

Factorize x2 – 2x – 8.

(b)

Write down the coordinates of each of these points

(i)

A;

(ii)

B.

Working:

Answers:

(a) …………………………………………..

(b) (i) ……………………………………..

(ii) ……………………………………..

(Total 4 marks)

2

3.

The diagram below shows a path x m wide around a rectangular lawn which measures 10 m by 8 m.

xm

8m

LAWN

xm

10 m

(a)

Write down an expression in terms of x for the area of the path.

(b)

What is the width of the path when its area is 208 m2?

Working:

Answers:

(a) …………………………………………..

(b) …………………………………………..

(Total 4 marks)

4.

The perimeter of a rectangle is 24 metres.

(a)

The table shows some of the possible dimensions of the rectangle. Find the values of a, b, c, d and e.

Length (m)

Width (m)

Area (m2)

1

11

11

a

10

b

3

c

27

4

d

e

(2)

3

(b)

If the length of the rectangle is x m, and the area is A m2, express A in terms of x only. (1)

(c)

What are the length and width of the rectangle if the area is to be a maximum? (3)

(Total 6 marks)

5.

(a)

Solve the equation x2 – 5x + 6 = 0.

(b)

Find the coordinates of the points where the graph of y = x2 – 5x + 6 intersects the x-axis.

Working:

Answers:

(a) …………………………………………..

(b) .................................................................. (Total 4 marks)

4

6.

A picture is in the shape of a square of side 5 cm. It is surrounded by a wooden frame of width x cm, as shown in the diagram below.

5 cm

x

l

The length of the wooden frame is l cm, and the area of the wooden frame is A cm2. (a)

Write an expression for the length l in terms of x.

(1)

(b)

Write an expression for the area A in terms of x.

(2)

(c)

If the area of the frame is 24 cm2, find the value of x.

(4)

(Total 7 marks)

5

7.

(a)

Factorize the expression 2x2 – 3x – 5.

(b)

Hence, or otherwise, solve the equation 2x2 – 3x = 5.

Working:

Answers:

(a) .................................................................. (b) .................................................................. (Total 4 marks)

8.

A rectangle has dimensions (5 + 2x) metres and (7 – 2x) metres. (a)

Show that the area, A, of the rectangle can be written as A = 35 + 4x – 4x2. (1)

(b)

The following is the table of values for the function A = 35 + 4x – 4x2. x

–3

–2

–1

0

1

2

3

4

A

–13

p

27

35

q

r

11

s

(i)

Calculate the values of p, q, r and s.

(ii)

On graph paper, using a scale of 1 cm for 1 unit on the x-axis and 1 cm for 5 units on the A-axis, plot the points from your table and join them up to form a smooth curve.

(6)

6

(c)

Answer the following, using your graph or otherwise.

(i)

Write down the equation of the axis of symmetry of the curve,

(ii)

Find one value of x for a rectangle whose area is 27 m2.

(iii)

Using this value of x, write down the dimensions of the rectangle. (4)

(d)

(i)

On the same graph, draw the line with equation A = 5x + 30.

(ii)

Hence or otherwise, solve the equation 4x2 + x – 5 = 0.

(3)

(Total 14 marks)

9.

(a)

Find the solution of the equation x2 – 5x – 24 = 0.

(b)

The equation ax2 – 9x – 30 = 0 has solution x = 5 and x = –2. Find the value of a.

Working:

Answers:

(a) ..............................................

(b)...

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