x plus a into x plus b

In this section, you will learn the formula or expansion for (x + a)(x + b).

That is, 

(x + a)(x + b)  =  x² + xb + xa + ab

(x + a)(x + b)  =  x² + (b + a)x + ab

(x + a)(x + b)  =  x² + (a + b)x + ab

Solved Problems

Problem 1 :

Find the product of : 

(x + 2)(x + 3)

Solution : 

(x + 2)(x + 3) is in the form of (x + a)(x + b)

Comparing (x + a)(x + b) and (x + 2)(x + 3), we get

x  =  x

a  =  2

b  =  3

Write the formula / expansion for (x + a)(x + b).

(x + a)(x + b)  =  x² + (a + b)x + ab

Substitute x for x, 2 for a and 3 for b. 

(x + 2)(x + 3)  =  x² + (2 + 3)x + (2)(3)

(x + 2)(x + 3)  =  x² + 5x + 6

So, the product of (x + 2) and (x + 3) is  

x² + 5x + 6

Problem 2 :

Find the product of : 

(2p + 1)(2p + 3)

Solution : 

(2p + 1)(2p + 3) is in the form of (x + a)(x + b)

Comparing (x + a)(x + b) and (2p + 1)(2p + 3), we get

x  =  2p

a  =  1

b  =  3

Write the formula / expansion for (x + a)(x + b).

(x + a)(x + b)  =  x² + (a + b)x + ab

Substitute 2p for x, 1 for a and 2 for b. 

(2p + 1)(2p + 3)  =  (2p)² + (1 + 3)(2p) + (1)(3)

(2p + 1)(2p + 3)  =  4p² + (4)(2p) + 3

(2p + 1)(2p + 3)  =  4p² + 8p + 3

So, the product of (2p + 1) and (2p + 3) is  

4p² + 8p + 3

x plus a times x minus b - Formula

(x + a)(x - b)  =  x² - xb + xa - ab

(x + a)(x - b)  =  x² + xa - xb - ab

(x + a)(x - b)  =  x² + (a - b)x - ab

x minus a times x plus b - Formula

(x - a)(x + b)  =  x² + xb - xa - ab

(x - a)(x + b)  =  x² - xa + xb - ab

(x - a)(x + b)  =  x² - (a - b)x - ab

x minus a times x minus b - Formula

(x - a)(x - b)  =  x² - xb - xa + ab

(x - a)(x - b)  =  x² - xa - xb + ab

(x - a)(x - b)  =  x² - (a + b)x + ab

Problem 3 :

4x² + mx + 9 = (2x + n)²

In the given equation m and n are the constants and n < 0. If the equation is true for all values of x, what is the value of m + n ?

Solution :

4x² + mx + 9 = (2x + n)²

4x² + mx + 9 = (2x)² + 2(2x)(n) + n²

4x² + mx + 9 = 4x² + 4xn + n²

mx + 9 = 4xn + n²

Comparing the corresponding terms, we get

4n = m and n² = 9 

From n² = 9, we get n = -3 and n = 3.

Since we have the condition n < 0, then n = -3

Applying the value of n in 4n = m, we get

4(-3) = m

m = -12

m + n = -12 + (-3) ==> -15

Problem 4 :

Which expression is equivalent to 2(x - 4)(2x + 3) ?

a) 4x² - 10x - 24     b) 4x² + 10x - 24

c) 4x² + 10x + 24       d) 8x² - 20x - 24

Solution :

2(x - 4)(2x + 3)

Using distributive property, multiplying the binomials, we get

= 2 (2x² + 3x - 8x - 12)

= 2 (2x² - 5x - 12)

By distributing 2, we get

= 4x² - 10x - 24

Problem 5 :

Which expression is equivalent to 16x⁴ - 81 ?

a) (4x² + 9)²     b) (2x + 3)²  (2x - 3)² 

c) (4x²  + 27) (4x²  - 3)      d) (4x² + 9)(2x + 3)(2x - 3)

Solution :

= 16x⁴ - 81

= 2⁴x⁴ - 3⁴

= (2x)⁴ - 3⁴

= ((2x)²))² - (3²)²

= ((2x)² + 3²)((2x)² - 3²)

= (4x² + 9)(2x + 3)(2x - 3)

Problem 6 :

Which expression is equivalent to

2(3x² - 5)² - (3x² - 5)²

a) 9x⁴ - 15x² + 75     b) 9x⁴ - 30x² + 25

c) 9x⁴ - 60x² + 75      d) 18x⁴ - 90x² + 25

Solution :

= 2(3x² - 5)² - (3x² - 5)²

= (3x² - 5)²

Using the formula for (a - b)²

= (3x²)²- 2(3x²)(5) + 5²

= 9x⁴- 30x² + 25

So, option b is correct.

Problem 7 :

Which expression is equivalent to

x²(x + 2)(x - 2) + 4 ?

a) (x² - 2)²       b) (x² + 2)²

c) (x - 1)² (x + 2)²       d) (x + 1)² (x - 2)²

Solution :

= x²(x + 2)(x - 2) + 4

= x²(x² - 4) + 4

= x⁴ - 4x² + 4

= (x²)²- 2x²(2) + 2²

= (x² - 2)²

So, option a is correct.

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