Maths

EXTENSION MATERIAL

Two more-challenging

quadratics

2

Try this worksheet after you have completed section 2.2

In Chapter 2 you learned different techniques for solving quadratic equations.

This worksheet looks at other equations that you can solve using the same techniques.

Solving by factorizing

EXAMPLE 1

Solve the equation x6 − 5x3 − 6 = 0 by factorizing

Answer

Remember, x 6 = ( x 3 )

Now this equation is similar to a quadratic

equation of the form ax 2 + bx + c = 0

( x 3 ) − 5( x 3 ) − 6 = 0

( x 3 + 1)( x 3 − 6) = 0

2

2

x3 1 0

x

3

x3

6 0

x

3

1

6

x

x

1

3

6

Using the zero-product property

EXAMPLE 2

Solve the equation (2x − 3)2 − 10(2x −3) + 9 = 0

Answer

This equation is also similar to a quadratic

equation of the form ax 2 + bx + c = 0

using the zero-product property

(2x − 3)2 − 10(2x − 3) + 9 = 0

((2x − 3) − 1)((2x − 3) − 9) = 0

( 2 x − 3) − 1 = 0 → 2 x − 3 = 1 → 2 x

=4 → x =2

2x − 3 − 9 = 0

2x − 12 = 0

2x = 12

x=6

Exercise 1

Solve these equations by factorizing.

1 x4 + 3x2 −10 = 0

3

x −4 x +3=0

5 (3x − 5)2 − 2(3x − 5) − 15 = 0

7

2 + x 2 1 + 2x 2

=

x 2 + 11 3 x 2 − 7

2 2x6 − 5x3 + 3 = 0

4 3x4 − 13x2 + 4 = 0

6

(

x − 5) + 5 ( x − 5) − 14 = 0

8

x3 + 3

x3 + 5

= 3

3

2 x − 4 3x − 9

2

© Oxford University Press 2012: this may be reproduced for class use solely for the purchaser’s institute

Extension worksheet

1

EXTENSION MATERIAL

Using the quadratic formula

EXAMPLE 3

Solve the equation x4 + 3x2 − 5 = 0 using the quadratic formula, giving the value of x

to 3 decimal places.

Check your answer using your GDC.

Answer

(x2 )

x2 =

Remember, x 4 = ( x 2 )

+ 3( x 2 ) − 5 = 0

2

−3 ±

(3 )

2

Solve for x2 using the quadratic formula,

with a = 1, b = 3 and c = −5.

− 4 (1) ( −5 )

2 (1)

x 2 = −3 +

Only use the positive value for x2

here, as you can’t take the square root of

a negative...