PROBLEMS ON QUADRATIC EQUATION

Factor :

Problem 1 :

x² + 6x + 9

Solution :

Product of factors = 9

Sum of factors = 6

Finding the factors of product and sum.

1 · 9 = 9 and 1 + 9 = 10 ≠ 6

3 · 3 = 9 and 3 + 3 = 6

So, the factors are 3 and 3

= x² + 3x + 3x + 9

By grouping,

= (x² + 3x) + (3x + 9)

By taking the common factor, we get

= x(x + 3) + 3(x + 3)

= (x + 3)(x + 3)

Problem 2 :

x² + 3x - 10

Solution :

Product of factors = -10

Sum of factors = 3

Finding the factors of product and sum.

(-1) · 10 = -10 and -1 + 10 = 9 ≠ 3

(-2) · 5 = -10 and -2 + 5 = 3

So, the factors are -2 and 5

= x² - 2x + 5x - 10

By grouping,

= (x² - 2x) + (5x - 10)

By taking the common factor, we get

= x(x - 2) + 5(x - 2)

= (x - 2) (x + 5)

Problem 3 :

What are the roots to the equation x² - 3x - 54 = 0?

a)  x = 6 or x = -9   b)  x = 1 or x = -54 

c)  x = 9 or x = -6  d)  x = 54 or x = -1

Solution :

x² - 3x - 54 = 0

By Splitting the middle term, we get

x² + 6x - 9x - 54 = 0

By taking the common factor, we get

x(x + 6) - 9(x + 6) = 0

(x - 9) (x + 6) = 0

x = 9 or x = -6

So, option (c) is correct.

Problem 4 :

Find the roots of x² - x - 12 = 0

Solution :

x² - x - 12 = 0

By Splitting the middle term, we get

x² + 3x - 4x - 12 = 0

By taking the common factor, we get

x(x + 3) - 4(x + 3) = 0

(x - 4) (x + 3) = 0

x = 4 or x = -3

So, the roots are x = 4 or x = -3.

Problem 5 :

Factor 3x² - x - 14

a)  (3x + 7) (x - 2)   b)  (x + 7) (3x - 2) 

 c)  (3x - 7) (x + 2)  d)  (x - 7) (3x + 2)

Solution :

= 3x² - x - 14

By Splitting the middle term, we get

= 3x² + 6x - 7x - 14

By taking the common factor, we get

= 3x(x + 2) - 7(x + 2)

= (3x - 7) (x + 2)

So, option (c) is correct.

Problem 6 :

Factor the expression 3x² + 15x + 18

Solution :

= 3x² + 15x + 18

Factoring 3,

= 3(x² + 5x + 6)

= 3(x² + 2x + 3x + 6)

By grouping,

= 3[(x² + 2x) + (3x + 6)]

= 3[x(x + 2) + 3(x + 2)]

= 3(x + 2) (x + 3)

Problem 7 :

What are the solutions of the equation m² = - 4m?

Solution :

m² = - 4m

m² + 4m = 0

(m + 0) (m + 4) = 0

m = 0 or m = -4

So, the solution is {0, -4}.