FACTORING QUADRATICS EXAMPLES

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Example 1 :

Factor :

x² + 13x + 30

Solution :

In the given quadratic expression, the coefficient of x² is 1.

Decompose the constant term +30 into two factors such that the product of the two factors is equal to +30 and the addition of two factors is equal to the coefficient of x, that is +13. 

Then, the two factors of +30 are 

+3 and +10

Factor the given quadratic expression using +3 and +10. 

x² + 13x +30  =  (x + 3)(x + 10)

Therefore, the factors of the given quadratic expression are

(x + 3) and (x + 10)

Example 2 :

Factor :

x² + 22x + 21

Solution :

In the given quadratic expression, the coefficient of x² is 1.

Decompose the constant term +21 into two factors such that the product of the two factors is equal to +21 and the addition of two factors is equal to the coefficient of x, that is 22. 

Then, the two factors of -21 are 

+1 and +21

Factor the given quadratic expression using +1 and +21. 

x² + 22x + 21  =  (x + 1)(x + 21)

Therefore, the factors of the given quadratic expression are

(x + 1) and (x + 21)

Example 3 :

Factor :

x² + 3x – 28

Solution :

In the given quadratic expression, the coefficient of x² is 1.

Decompose the constant term -28 into two factors such that the product of the two factors is equal to -28 and the addition of two factors is equal to the coefficient of x, that is +3. 

Then, the two factors of -28 are 

-4 and +7

Factor the given quadratic expression using -4 and +7. 

x² + 3x – 28  =  (x - 4)(x + 7)

Therefore, the factors of the given quadratic expression are

(x - 4) and (x + 7)

Example 4 :

Factor :

x² – x – 30

Solution :

In the given quadratic expression, the coefficient of x² is 1.

Decompose the constant term -30 into two factors such that the product of the two factors is equal to -30 and the addition of two factors is equal to the coefficient of x, that is -1. 

Then, the two factors of -30 are 

-6 and +5

Factor the given quadratic expression using -6 and +5. 

x² – x – 30  =  (x - 6)(x + 5)

Therefore, the factors of the given quadratic expression are

(x - 6) and (x + 5)

Example 5 :

Factor :

x² – 2x – 24

Solution :

= x² – 2x – 24

= x² – 6x + 4x – =24

= x(x - 6) + 4(x - 6)

= (x - 6)(x + 4)

Example 6 :

Factor :

3x² + 7x – 6

Solution :

= 3x² + 7x – 6

= 3x² + 9x - 2x – 6

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

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

Example 7 :

Factor :

-x² + 5x + 84

Solution :

= -x² + 5x + 84

= -x² + 12x - 7x + 84

= -x(x - 12) - 7(x - 12)

= (-x - 7)(x - 12)

Example 8 :

Factor :

-6x² + x + 2

Solution :

= -6x² + x + 2

= -6x² + 4x - 3x + 2

= -2x(3x - 2) - 1(3x - 2)

= (-2x - 1)(3x - 2)

Example 9 :

3x² -  48

Which of the following is equivalent to the expression above.

a)  3(x - 4)(x + 4)     b)  3(x - 4)²    c)  (3x - 4)(3x + 4)

d)  (3x + 4)(x - 4)

Solution :

= 3x² -  48

Factoring 3, we get

= 3(x² -  16)

= 3(x² - 4²)

Using algebraic identity a² - b², we get (a + b) (a - b)

= 3(x + 4)(x - 4)

So, option a is correct.

Example 10 :

If x + y = 10 and x - y = 4, what is the value of x² - y² ?

a)  20     b)  24   c)  36   d)  40

Solution :

x² - y² = (x + y)(x - y)

Given values are x + y = 10 and x - y = 4

x² - y² = 10(4)

= 40

so, option d is correct.

Example 11 :

6x² + 7x - 24 = 0

If r and s are two solutions of the equation above and r > s, which of the following is the value of r - s ?

a)  7/6     b)  16/3   c)  25/6   d)  20/3

Solution :

6x² + 7x - 24 = 0

Solving this quadratic equation using factoring method,

6x² + 16x - 9x - 24 = 0

2x(3x + 8) - 3(3x + 8) = 0

(2x - 3)(3x + 8) = 0

Equating each factor to 0, we get

So, the solutions are 3/2 and -8/3

r = 3/2 > s = -8/3

r - s = 3/2 - (-8/3)

= 3/2 + (8/3)

= (9 + 16)/6

= 25/6

So, option c is correct.

Example 12 :

x² - 3x = 28

If r and s are two solutions of the equation above, which of the following is the value of r + s ?

a)  -3    b)  3   c)  6   d)  9

Solution :

x² - 3x = 28

x² - 3x - 28 = 0

x² + 7x - 4x - 28 = 0

x(x + 7) - 4(x + 7) = 0

(x - 4)(x + 7) = 0

Equating each factor to 0, we get

x - 4 = 0 and x + 7 = 0

x = 4 and x = 7

r =4 and s = 7

r + s = 4 + 7

= 11

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