104 Basic Engineering Mathematics
Hence, (2x + 3)(2x + 1) = 0, from which either
(2x + 3) = 0 or (2x + 1) = 0.
Thus,
2x = −3, from which x = −
3
2
or −1.5
or
2x = −1, from which x = −
1
2
or −0.5
which may be checked in the original equation.
Problem 11. Solve the quadratic equation
15x 2 + 2x − 8 = 0 by factorizing
The factors of 15x 2 are 15x and x or 5x and 3x.
The factors of −8 are −4 are +2, or 4 and −2, or −8
and +1, or 8 and −1.
By trial and error the only combination that works is
15x
2
+ 2x − 8 = (5x + 4)(3x − 2)
Hence, (5x + 4)(3x − 2) = 0, from which either 5x +
4 = 0 or 3x − 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 in x
If the roots of a quadratic equation are, say, α and β,
then (x − α)(x − β) = 0.
Hence, if α =
1
3
and β = −2,
x −
1
3
(x − (−2)) = 0
x −
1
3
(x + 2) = 0
x
2
−
1
3
x + 2x −
2
3
= 0
x
2
+
5
3
x −
2
3
= 0
or
3x
2
+ 5x − 2 = 0
Problem 13. Find the equation in x whose 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
− 5x + 5x − 25 = 0
i.e.
x
2
− 25 = 0
Problem 14. Find the equation in x whose 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
− 1.2x + 0.4x − 0.48 = 0
i.e.
x
2
− 0.8x − 0.48 = 0
Now try the following Practice Exercise
Practice Exercise 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 = 0
2 . x 2 + 4x − 32 = 0
3. (x + 2) 2 = 16
4. 4x 2 − 9 = 0
5. 3x 2 + 4x = 0
6 . 8 x 2 − 32 = 0
7. x 2 − 8x + 16 = 0
8. x 2 + 10x + 25 = 0
9. x 2 − 2x + 1 = 0
10. x 2 + 5x + 6 = 0
11. x 2 + 10x + 21 = 0 12. x 2 − x − 2 = 0
13. y 2 − y − 12 = 0
14. y 2 − 9y + 14 = 0
15. x
2
+ 8x + 16 = 0
16. x
2
− 4x + 4 = 0
17. x 2 + 6x + 9 = 0
18. x 2 − 9 = 0
19. 3x 2 + 8x + 4 = 0
20. 4x 2 + 12x + 9 = 0
21. 4z 2 −
1
16
= 0
22. x 2 + 3x − 28 = 0
23. 2x 2 − x − 3 = 0
24. 6x 2 − 5x + 1 = 0
25. 10x 2 + 3x − 4 = 0 26. 21x 2 − 25x = 4
27. 8x 2 + 13x − 6 = 0 28. 5x 2 + 13x − 6 = 0
29. 6x 2 − 5x − 4 = 0
30. 8x 2 + 2x − 15 = 0
In problems 31 to 36, determine the quadratic
equations in x whose roots are
31. 3 and 1
32. 2 and −5
33. −1 and −4
34. 2.5 and −0.5
35. 6 and −6
36. 2.4 and −0.7
Hence, (2x + 3)(2x + 1) = 0, from which either
(2x + 3) = 0 or (2x + 1) = 0.
Thus,
2x = −3, from which x = −
3
2
or −1.5
or
2x = −1, from which x = −
1
2
or −0.5
which may be checked in the original equation.
Problem 11. Solve the quadratic equation
15x 2 + 2x − 8 = 0 by factorizing
The factors of 15x 2 are 15x and x or 5x and 3x.
The factors of −8 are −4 are +2, or 4 and −2, or −8
and +1, or 8 and −1.
By trial and error the only combination that works is
15x
2
+ 2x − 8 = (5x + 4)(3x − 2)
Hence, (5x + 4)(3x − 2) = 0, from which either 5x +
4 = 0 or 3x − 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 in x
If the roots of a quadratic equation are, say, α and β,
then (x − α)(x − β) = 0.
Hence, if α =
1
3
and β = −2,
x −
1
3
(x − (−2)) = 0
x −
1
3
(x + 2) = 0
x
2
−
1
3
x + 2x −
2
3
= 0
x
2
+
5
3
x −
2
3
= 0
or
3x
2
+ 5x − 2 = 0
Problem 13. Find the equation in x whose 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
− 5x + 5x − 25 = 0
i.e.
x
2
− 25 = 0
Problem 14. Find the equation in x whose 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
− 1.2x + 0.4x − 0.48 = 0
i.e.
x
2
− 0.8x − 0.48 = 0
Now try the following Practice Exercise
Practice Exercise 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 = 0
2 . x 2 + 4x − 32 = 0
3. (x + 2) 2 = 16
4. 4x 2 − 9 = 0
5. 3x 2 + 4x = 0
6 . 8 x 2 − 32 = 0
7. x 2 − 8x + 16 = 0
8. x 2 + 10x + 25 = 0
9. x 2 − 2x + 1 = 0
10. x 2 + 5x + 6 = 0
11. x 2 + 10x + 21 = 0 12. x 2 − x − 2 = 0
13. y 2 − y − 12 = 0
14. y 2 − 9y + 14 = 0
15. x
2
+ 8x + 16 = 0
16. x
2
− 4x + 4 = 0
17. x 2 + 6x + 9 = 0
18. x 2 − 9 = 0
19. 3x 2 + 8x + 4 = 0
20. 4x 2 + 12x + 9 = 0
21. 4z 2 −
1
16
= 0
22. x 2 + 3x − 28 = 0
23. 2x 2 − x − 3 = 0
24. 6x 2 − 5x + 1 = 0
25. 10x 2 + 3x − 4 = 0 26. 21x 2 − 25x = 4
27. 8x 2 + 13x − 6 = 0 28. 5x 2 + 13x − 6 = 0
29. 6x 2 − 5x − 4 = 0
30. 8x 2 + 2x − 15 = 0
In problems 31 to 36, determine the quadratic
equations in x whose roots are
31. 3 and 1
32. 2 and −5
33. −1 and −4
34. 2.5 and −0.5
35. 6 and −6
36. 2.4 and −0.7