According to the binomial theorem, the sum and product of the constant and variable terms will be a quadratic equation. The expansion of (x ± a)(x ± b) is the individual product of the variable x and constant terms. It can be read as the square of x and the sum of the constant terms and product of the constant terms. Read the entire article to derive (x + a)(x + b), (x – a)(x – b), (x + a)(x – b) and (x – a)(x + b). In addition to the derivations, the students can find examples of Expansion of Powers of Binomials and Trinomials.

Expansion of (x ± a)(x ± b)

Let us discuss briefly the expansion of (x ± a)(x ± b) with proves from here.

What is the expansion of (x + a)(x + b)?
The derivation of expanding (x + a)(x + b) is given below,
(x + a)(x + b) = x(x + b) + a(x + b)
(x + a)(x + b) = x² + bx + ax + ab
(x + a)(x + b) = x² + x(a + b) + ab

What is the expansion of (x – a)(x – b)?
The derivation of expanding (x – a)(x – b) is given below,
(x – a)(x – b) = x(x – b) – a(x – b)
(x – a)(x – b) = x² – bx – ax + ab
(x – a)(x – b) = x² – x(a + b) + ab

What is the expansion of (x + a)(x – b)?
Derivation on how to expand (x + a) (x – b) is as follows,
(x + a)(x – b) = x(x – b) + a(x – b)
(x + a)(x – b) = x² – bx + ax – ab
(x + a)(x – b) = x² + x(a – b) – ab

What is the expansion of (x – a)(x + b)?
The derivation of expanding (x – a)(x – b) is given below,
(x – a)(x + b) = x(x + b) – a(x + b)
(x – a)(x + b) = x² + bx – ax – ab
(x – a)(x + b) = x² – x(a – b) – ab

Formulas of (x ± a)(x ± b)

  1. (x – a)(x + b) = x² – x(a – b) – ab
  2. (x + a)(x – b) = x² + x(a – b) – ab
  3. (x – a)(x – b) = x² – x(a + b) + ab
  4. (x + a)(x + b) = x² + x(a + b) + ab

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Solved Problems on Expansion of (x ± a)(x ± b)

Example 1.
Find the product of (x + 2) (x + 4) using the standard formula.
Solution:
Given the binomial expression (x + 2) (x + 4)
By using the formula (x + a)(x + b) = x² + x(a + b) + ab we can expand the given expression.
(x + 2) (x + 4) = x² + x(2 + 4) + (2)(4)
(x + 2) (x + 4) = x² + 6x+ 8
Thus the product of (x + 2) (x + 4) is x² + 6x+ 8

Example 2.
Expand (a – 5)(a – 10) using the (x ± a)(x ± b) formula.
Solution:
Given the binomial expression (a – 5)(a – 10)
By using the formula (x – a)(x – b) = x² – x(a + b) + ab
(a – 5)(a – 10) = a² – a(5 + 10) + 5(10)
(a – 5)(a – 10) = a² – 15a + 50
Thus the expansion of (a – 5)(a – 10) is a² – 15a + 50

Example 3.
Find the product of (m + 5)(m + 1) using the standard formula.
Solution:
Given,
(m + 5)(m + 1)
By using the formula (x + a)(x + b) = x² + x(a + b) + ab we can expand the given expression.
(m + 5)(m + 1) = m² + m(5 + 1) + 5
(m + 5)(m + 1) = m² + m(6) + 5
(m + 5)(m + 1) = m² + 6m + 5
Thus the expansion of (m + 5)(m + 1) is m² + 6m + 5

Example 4.
Find the product of (x + 2)(x – 3) using the standard formula.
Solution:
Given the binomial expression (x + 2)(x – 3)
By using the formula (x + a)(x – b) = x² + x(a – b) – ab we can expand the given expression.
(x + 2)(x – 3) = x² + x(2 – 3) – 2(3)
(x + 2)(x – 3) = x² + x(-1) – 2(3)
(x + 2)(x – 3) = x² – x – 6

Example 5.
Find the product of (2m + 1)(2m – 4) using the standard formula.
Solution:
Given the binomial expression (2m + 1)(2m – 4)
By using the formula (x + a)(x – b) = x² + x(a – b) – ab we can expand the given expression.
(2m + 1)(2m – 4) = (2m)² + 2m(1 – 4) – 1(4)
(2m + 1)(2m – 4) = 4m² + 2m – 8m – 4
(2m + 1)(2m – 4) = 4m² – 6m – 4

FAQs on Expansion of (x ± a)(x ± b)

1. What is the expansion of (x + a)(x + b)?
The expansion of (x + a)(x + b) is x² + x(a + b) + ab

2. What is the expansion of (x + a)(x – b)?
The expansion of (x + a)(x – b) is x² + x(a – b) – ab

3. How to expand (x + a)(x + b)?
(x + a)(x + b) = x(x + b) + a(x + b)
(x + a)(x + b) = x² + bx + ax + ab
(x + a)(x + b) = x² + x(a + b) + ab

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