Quadratics(Identities)

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© P. Kavanagh, 2006

Type One• Multiply out both sides of the equation.• Equate matching coefficients.• You may need to solve a set of simultaneous

equations.Type Two• Solve one equation.• The coefficients of the next equation will be equal

to one just solved.• Put the solutions of the last equation equal to the

new unknown expression.

Identity Equations

© P. Kavanagh, 2006 1995 Q1 (a)

2 2

2 2

2

If 2x 5x 6 p(x q) r for all x, find the value of p, of q and of r.

Square out the right hand side and multiply it all out completely and

then match coefficients on both sides.

2x 5x 6 p(x q) r

2x 5x

2

2

2

2

2

2

2

6 p[x 2xq q ] r

Matching up

p 2 2pq 5 pq r 62(2)q 5 5

2 r 65 4q4 25 23

r 68 8

5 23

5x 2x6 pq r

p 2; q and r

p qx px

4

2

8

© P. Kavanagh, 2006 2001 Q1 (a)

2 2

2

2 2

2

2

2

Find the real numbers a and b such that x 4x 6 (x a) b for all x .

x 4x 6 (x a) b

Matching up

2a 4 a b 6

4 6x 2xax x

a 2 4

b

a

1

b

b 6

0

¡

© P. Kavanagh, 2006 1997 Q2 (b)

2

2

2

2

Solve x 6x 8 0, and hence find the values of x for which

1 1x 6 x 8 0

x x

x 6x 8 0

(x 4)(x 2) 0

x 4 or x 2

The hence means that the previous part should help us with this part

1 1x 6 x

x x

2 2

2 2

8 0

Same coefficients as one solved above, so put the previous solutions

1equal to x .

x1 1

x 4 or x 2x x

x 1 4x x 1 2x

x 4x 1 0 x 2x 1 0

(x 1)(x 1) 04 16 4x

x 12

x 2 3

© P. Kavanagh, 2006 2000 Q2 (b)

2

2

2

2

Solve x 2x 24 0.

Hence find the values of x for which

4 4x 2 x 24 0, x , x 0.

x x

x 2x 24 0

(x 6)(x 4) 0

x 6 or x 4

4 4Solving x 2 x 24 0

x x

Same coefficients as one solved above

x

¡

2 2

2 2

2

4 46 or x 4

x xx 4 6x x 4 4x

x 6x 4 0 x 4x 4 0

(x 2)(x 2) 0( 6) ( 6) 4(1)(4)x x 22

x 3 5

Quadratics(Alpha & Beta, Nature of Roots)

2

2 2

b b 4acx

2a

b cA quadratic equation takes the form ax bx c 0 i.e. x x 0 where the:

a a

bsum of the roots:

a

product of the

Roots of a Quadratic Equation:

Sum and Product of a Quadratic Equation:

2 2 2

3 3 2 2

3 3 2 2

2

c roots:

a

( ) 2

( )( )

( )( )

x (sum of roots)x product of the roots 0

Identities to be familiar with:

Forming an equation:

Nature of the roots of a Quadratic Equ2

2

2

b 4ac for real roots

b 4ac 0 for unreal/complex roots

b 4ac 0 for equal roots

ation:

Quadra

tic Equatio

n

Facts

© P. Kavanagh, 2006

Roots of a Quadratic

2

2

2Solve x 3x 2 0

bx

2a

3x

2x 2 and x 1

as

we have real distin

b 4

ct

ac

1

b 4a

ro

c 0

ots.

2

2

2Solve x 4x 4 0

bx

2a

4x

2x 2 and x 2

as

we have real equ

b 4ac

0

b 4ac 0

al roots.

2

2

2Solve x 6x 13 0

bx

2a

6x

2

16 i

b 4ac

16

b 4ac 0

s not real

as

we have no real roots.

© P. Kavanagh, 2006

Related Roots

and 1The roots of the equation are consectutive integers.

orThe roots of the equations differ by one.

and 1

One root of the equation is twice the other. and 2

The roots of the equation differ by

and 2

two. or

and 2

If there is a relationship between the roots given in the question, then

instead of using and , express both of the roots in terms of . (or !)

© P. Kavanagh, 2006 1994 Q2 (b)

2

2 2

2

2

2

The roots of the equation 2x 6x 3 0 are and .

Show that 6.

The roots of 2x px q 0 are 2 and 2 .

Find the value of p and the value of q.

2x 6x 3 0

3[N.Bx 3x . Vital Step!]

Sum Produc2

t

0

2 2 2

2 2 2

3 32

( ) 2

3( 3) 2 9 3 6

2

© P. Kavanagh, 2006 1994 Q2 (b) ctd.

2

2 2

2

2

2

The roots of the equation 2x 6x 3 0 are and .

Show that 6.

The roots of 2x px q 0 are 2 and 2 .

Find the value of p and the value of q.

2x px q 0

p qx x 0

2[N.B. Vital Step!]

Sum Produ2

ct

2

2 2

2 2

p q2 (2 )( 2 )

2 2p q

3 3 2 5 22 2p q

3( ) 2( ) 52 2

(2)(3)( 3) p 3 q2(6) 5

2 218 p

39 q

© P. Kavanagh, 2006 1995 Q2 (b) (i)

2 2 2

2

2 2 2

2 2 2

2 2

2 2

2 2

Sum Produ

If and are the roots of the equation x px q 0, show that ( ) p 4q.

x px q 0

p q

( ) 2

( ) 2

( ) ( ) 2 2

( ) p 2q 2q

( ) p

c

4

t

q

© P. Kavanagh, 2006 1996 Q2 (c)

1 1Find the quadratic equation with roots and given that 5 and k, where k 0.

Find the range of values of k for which the equation will ha

Sum Product

New sum New produ

ve real roots.

5 k

1

ct

2

2

2

2

1 1 1

5 1 1k k

x (Sum)x Product 0

5 1x x 0

k kkx

If the roots are re

5

a

x 1 0

25 4k 0

2

l b 4ac

5k

0

4

© P. Kavanagh, 2006 1998 Q1 (c) (part of)

2

2

2

2

If the quadratic equation ax bx c 0 has equal roots, solve for x in terms of a an

Equal roo

d b,

where a,

t

b, c .

ax

s b

bx c 0

b b 4acx

2a

b 0 bx

4ac 0

2a 2a

¡

© P. Kavanagh, 2006 1998 Q2 (b)

2

3 3

3 3

2

3 3 2 2

3 3 2 2 2 2 2

If and are the roots of the equation x 6x 2 0, 0, 0, find and .

Factorise .

Find the

Sum Produ

value of .

x 6x 2 0

6 2

( )( )

( )( ) ( )

ct

2

3 3 2 2 2

3 3

(6)(32 2) (6) 2(2) 32

(6)(30) 180

© P. Kavanagh, 2006 2001 Q2 (c)

2 2

2

and are real numbers such that 7 and 11.

Show that 27

Find a quadratic equation with roots and and write your answer in the

form px qx r 0 where p, q, r .

Sum Product

(i)

(ii)

(i)

¢

2 2 2

2 2 2

2 2

2

2

2

New Sum New Produ

7 11

( ) 2

( 7) 2(11) 49 22 27

1

2711

x (Sum)x Product 0

27x x 1 0

1111x 27x 1

c

0

t

1

(ii)

© P. Kavanagh, 2006 2002 Q1 (c) (i)

2

2

(p r t)x 2rx (t r p) 0 is a quadratic equation, where p, r and t are integers.

Show that

the roots are rational

one of the roots is an intege

b 4ac is a perfect square if the roots ar

r

e ra

(i)

(ii)

Hint :

2

2

2

2 2 2

2 2

2

(p r t)x 2rx (t r p) 0

(2r) 4(p r t)(t r p)

4r 4

b 4ac is

[r (p t)][r

a perfect square

(p t)] [rearra

if the roots ar

nging so it is the diff. of 2 sqs.]

4r 4[r (p t) ]

4r 4r 4(p

e ratio

tiona

a

.

n l.

l

(i)

2 2t) 4(p t)

This is a perfect square so the roots are rational.

Rational: the answer can be expressed as a fraction.

Perfect Square: a perfect square has a whole number answer for its root, e.g. 9, 25, 625 are perfect squares whereas 7, 13, 110 are not.

© P. Kavanagh, 2006 2002 Q1 (c) (ii)

2

2

2

(p r t)x 2rx (t r p) 0 is a quadratic equation, where p, r and t are integers.

Show that

the roots are rational

one of the roots is an integer

b b 4acFind the roots using x

2a

2r 4(p t)x

2

(i)

(ii)

(ii)

(p r t)

2r (2p 2t)x

2p 2r 2t

2r 2p 2t 2r (2p 2t)x or x

2p 2r 2t 2p 2r 2t

2r 2p 2t 2r 2p 2tx or x

2p 2r 2t 2p 2r 2t

2(p r t) 2(p r t)x or x

2(p r t) 2(p r t)

p r tx or x 1

p r t

Note: Parts (i) and (ii) could both have been answered using this as it can be shown that in fact both of the roots are rational.

© P. Kavanagh, 2006 2002 Q2 (c) (i)

2 2

2

2

2

2

2

Show that if the roots of x bx c 0 differ by 1, then b 4ac 1.

Let the roots be and 1

1 b ( 1) c

2 1 b cb 12

c

b 1 b 1c

2 2

b 2b 1 (b 1)c

4 2b 2b 1

Sum Prod

2b

uc

b

t

2 4c

2 4c 1

© P. Kavanagh, 2006 2002 Q2 (c) (ii)

2

2

The roots of the eqaution x (4k 5)x k 0 are consecutive integers.

Using the result from part (i), or otherwise, find the value of k and the roots of the equation

b 4c 1 as the roots are consecutive

2

2 2

2

2

integers [from part (i)]

(4k 5) 4k 1

16k 40k 25 4k 1 0 x 3x 2 0

16k 44k 24 0 (x 1)(x 2) 0

4k 11k 6 0 x 1 or x 2

(4k 3)(k 2) 0

3k or k 2

4k 2 as k is an integer.

© P. Kavanagh, 2006 2003 Q1 (b) (ii)

2

2

2

22

2

Show that 2x 3 is a factor of 4x 2(1 3)x 3 and find the other factor.

2x 14x 2(1 3)x 32x 3 4x 2(1 3)x 3(2x 3)(2x 1)

4x 2x 34x 2x 2x 3 3

2x 34x 2x(1 3) 3

Hence (2x 3) and (2x 1) are both

factors.

or

2x 3

0

© P. Kavanagh, 2006 2003 Q1 (c)

2

2

The real roots of x 10x c 0 differ by 2p where c, p and p 0.

Show that p 25 c

Given that one root is greater than zero and the other root is less than zero,

find the range of possible values

(i)

(ii)

¡

2

2

2

2 2

2

2

of p.

Roots differ by 2p Make sure the the big

Let the roots be be and 2p

x 10x c 0

2p 10 ( 2p) c

2 2p 10 2p c2p 10

p 5

(p 5) 2p(p 5) c

p 10p 25

Sum Produc

2p 10p c

p 25 c

p 25 c

t

(i) (ii) ger root

is greater than zero.

0 2p 0

p 5 0 p 5 2p 0

p 5 p 5

p 5

as p 0,p 5

© P. Kavanagh, 2006 2003 Q2 (c) (ii)

2 2 2 2

2 2

2 2 2 2

2 22

2

Given that x and x are the solutions of the quadratic equation

2k x 2ktx t 3k 0, where k, t and k 0,

show that is independent of k and t.

2k x 2ktx t 3k 0

t t 3kx x 0

Sum Product

k 2k

¡

2 2

2

2 2 2

2 2 22 2

2

2 2 22 2

2 2

2 22 2

2 2

2 2

t t 3kk 2k

( ) 2

t t 3k2

k 2k

t t 3kk k

t t3

k k3

© P. Kavanagh, 2006 2004 Q2 (b) (ii)

2

2 2

2

2 2 2 2

3 3 3

3 3

2

The roots of x px q 0 are and , where p,q

Find the quadratic equation whose roots are and .

x px q 0

p q

( )q

( p)(q)qpq

x (

Sum Product

New Sum New Product

¡ .

2 3

2 3

Sum)x Product 0

x ( pq)x q 0

x pq q 0

© P. Kavanagh, 2006 2004 Q2 (c) (i)

2 2

2

2 2

2 2

2 2

2 2

2

Show that for any real values of a, b and h, the quadratic equation

(x a)(x b) h 0

has real roots.

(x a)(x b) h 0

x bx ax ab h 0

x (a b)x ab h 0

(a b) 4(1)(ab h )

(a

Real roots b 4

b) 4

ac

a

0

b 4h

2 2 2

2 2 2

2 2

a 2ab b 4ab 4h

a 2ab b 4h

(a b) 4h

0

© P. Kavanagh, 2006

089

1

2

Testing

9( ) 8( ) 0

True

1 1

2006 Q2 (b)

2

2

2

Find the range of values of t for which the quadratic equation

(2t 1)x 5tx 2t 0 has real roots.

Explain why the roots are real when t is an

Real root

integer.

(5t) 4(2t 1)

s b

(2t) 0

0

2

4ac

(i)

(ii)

(i)

¡

2 2

2

5t 16t 8t 0

9t 8t 0

t(9t 8) 0

t 0 or 9t 8 0

8t

98

t and t 09

8There is no integer value for t 0.

9Hence all integer roots are real.

(ii)