Basic Engineering Mathematics, Fifth Edition

104 Basic Engineering Mathematics

Hence, ( 2 x+ 3 )( 2 x+ 1 )=0, from which either

( 2 x+ 3 )=0or( 2 x+ 1 )=0.

Thus, 2x=− 3 ,from which x=−

3

2

or − 1. 5

or 2 x=− 1 ,from which x=−

1

2

or − 0. 5

which may be checked in the original equation.

Problem 11. Solve the quadratic equation

15 x^2 + 2 x− 8 =0 by factorizing

The factors of 15x^2 are 15xandxor 5xand 3x.

The factors of−8are−4are+2, or 4 and−2, or− 8

and+1, or 8 and−1.

By trial and error the only combination that works is

15 x^2 + 2 x− 8 =( 5 x+ 4 )( 3 x− 2 )

Hence,( 5 x+ 4 )( 3 x− 2 )=0, from which either 5x+

4 =0or3x− 2 =0.

Hence,x=−

4

5

or x=

2

3

which may be checked in the original equation.

Problem 12. The roots of a quadratic equation

are

1

3

and−2. Determine the equation inx

If the roots of a quadratic equation are, say,αandβ,

then(x−α)(x−β)=0.

Hence, ifα=

1

3

andβ=−2,

(

x−

1

3

)

(x−(− 2 ))= 0

(

x−

1

3

)

(x+ 2 )= 0

x^2 −

1

3

x+ 2 x−

2

3

= 0

x^2 +

5

3

x−

2

3

= 0

or 3 x^2 + 5 x− 2 = 0

Problem 13. Find the equation inxwhose roots

are 5 and− 5

If 5 and−5 are the roots of a quadratic equation then

(x− 5 )(x+ 5 )= 0

i.e. x^2 −^5 x+^5 x−^25 =^0

i.e. x^2 −^25 =^0

Problem 14. Find the equation inxwhose roots

are 1.2 and− 0. 4

If 1.2 and− 0 .4 are the roots of a quadratic equation

then

(x− 1. 2 )(x+ 0. 4 )= 0

i.e. x^2 − 1. 2 x+ 0. 4 x− 0. 48 = 0

i.e. x^2 − 0. 8 x− 0. 48 = 0

Now try the following Practice Exercise

PracticeExercise 54 Solving quadratic

equations by factorization (answers on

page 346)

In problems 1 to 30, solve the given equations by

factorization.

1. x^2 − 16 =02.x^2 + 4 x− 32 = 0


2. (x+ 2 )^2 = 16 4. 4x^2 − 9 = 0


3. 3x^2 + 4 x=06.8x^2 − 32 = 0


4. x^2 − 8 x+ 16 =08.x^2 + 10 x+ 25 = 0


5. x^2 − 2 x+ 1 = 0 10. x^2 + 5 x+ 6 = 0


6. x^2 + 10 x+ 21 =0 12. x^2 −x− 2 = 0


7. y^2 −y− 12 = 0 14. y^2 − 9 y+ 14 = 0


8. x^2 + 8 x+ 16 =0 16. x^2 − 4 x+ 4 = 0


9. x^2 + 6 x+ 9 = 0 18. x^2 − 9 = 0


10. 3x^2 + 8 x+ 4 =0 20. 4x^2 + 12 x+ 9 = 0


11. 4z^2 −

1

16

= 0 22. x^2 + 3 x− 28 = 0

23. 2x^2 −x− 3 = 0 24. 6x^2 − 5 x+ 1 = 0


24. 10x^2 + 3 x− 4 =0 26. 21x^2 − 25 x= 4


25. 8x^2 + 13 x− 6 =0 28. 5x^2 + 13 x− 6 = 0


26. 6x^2 − 5 x− 4 =0 30. 8x^2 + 2 x− 15 = 0

In problems 31 to 36, determine the quadratic

equations inxwhose roots are

31. 3 and 1 32. 2 and− 5


32. −1and− 4 34. 2.5 and− 0. 5


33. 6 and− 6 36. 2.4and− 0. 7