Quadratics: square form, discriminants and roots: practice

7 marksCore techniqueP2.3

The curve

C

has equation

y = 2x^2 - 12x + 13

.

(a)

Write

2x^2 - 12x + 13

in the form

a(x + b)^2 + c

, where

a

,

b

and

c

are constants to be found.

(3 marks)

(b)

Hence write down the coordinates of the minimum point of

C

.

(2 marks)

(c)

The line

y = k

, where

k

is a constant, meets

C

at two distinct points. Find the set of possible values of

k

.

(2 marks)

6 marksCore techniqueP2.3

The equation

(k + 1)x^2 + 4x + (k - 2) = 0

, where

k

is a constant, has no real roots.

(a)

Show that

k^2 - k - 6 > 0

.

(3 marks)

(b)

Hence find the set of possible values of

k

.

(3 marks)

6 marksUnfamiliar problemP2.3

The curve

C

has equation

y = x^2 + (p - 3)x + p

, where

p

is a constant. The minimum point of

C

lies on the

x

-axis.

(a)

Find the two possible values of

p

.

(3 marks)

(b)

Given also that the minimum point of

C

lies to the right of the

y

-axis, find

p

and the coordinates of the minimum point.

(3 marks)

7 marksUnfamiliar problemP2.3, OT3.3

A stone is thrown upwards from the top of a cliff. Its height above the sea,

h

metres,

t

seconds after it is thrown is modelled by

h = 20 + 15t - 5t^2

.

(a)

Write

20 + 15t - 5t^2

in the form

A - B(t - C)^2

, where

A

,

B

and

C

are positive constants.

(3 marks)

(bi)

Hence state the maximum height of the stone above the sea.

(1 mark)

(bii)

State the time at which the maximum height occurs.

(1 mark)

(c)

Find the time at which the model predicts the stone reaches the sea.

(2 marks)

4 marksCore techniqueShow full workingP2.3, P2.1

Solutions relying entirely on calculator technology are not acceptable.

Solve

2^{2x+1} - 17 \times 2^{x} + 8 = 0

.

(4 marks)

4 marksCore techniqueP2.3

f(x)=2x^2-12x+21

.

(a)

Express

f(x)

in completed-square form. Give exact answers.

(2 marks)

(b)

Determine the values of

c

for which

f(x)=c

has two distinct positive roots.

(2 marks)

4 marksCore techniqueP2.3

f(x)=2x^2-16x+35

.

(a)

Express

f(x)

in completed-square form. Give exact answers.

(2 marks)

(b)

Determine the values of

c

for which

f(x)=c

has two distinct positive roots.

(2 marks)

4 marksUnfamiliar problemP2.3

A line

y=mx+9

meets the parabola

y=x^2

.

(a)

Show that there are two distinct intersections for every real

m

.

(2 marks)

(b)

Given that one intersection has

x=6

, find the other x-coordinate. Give exact answers.

(2 marks)

4 marksUnfamiliar problemP2.3

A line

y=mx+16

meets the parabola

y=x^2

.

(a)

Show that there are two distinct intersections for every real

m

.

(2 marks)

(b)

Given that one intersection has

x=8

, find the other x-coordinate. Give exact answers.

(2 marks)

5 marksUnfamiliar problemP2.3

A quadratic is

x^2-2tx+3t+9=0

.

(a)

Find all values of

t

for which the quadratic has equal roots. Give exact answers.

(3 marks)

(b)

Deduce the repeated root in terms of

t

. Give exact answers.

(2 marks)

5 marksUnfamiliar problemP2.3

A quadratic is

x^2-2tx+4t+16=0

.

(a)

Find all values of

t

for which the quadratic has equal roots. Give exact answers.

(3 marks)

(b)

Deduce the repeated root in terms of

t

. Give exact answers.

(2 marks)