Expansion of (x ± a)(x ± b)

We will discuss here about the expansion of (x ± a)(x ± b)

(x + a)(x + b) = x(x + b) + a (x + b)

                      = x\(^{2}\) + xb + ax + ab

                      = x\(^{2}\) + (b + a)x + ab

(x - a)(x - b) = x(x - b) - a (x - b)

                    = x\(^{2}\) - xb - ax + ab

                    = x\(^{2}\) - (b + a)x + ab

(x + a)(x - b) = x(x - b) + a (x - b)

                     = x\(^{2}\) - xb + ax - ab

                     = x\(^{2}\) + (a - b)x - ab

(x - a)(x + b) = x(x + b) - a (x + b)

                     = x\(^{2}\) + xb - ax - ab

                     = x\(^{2}\) - (a - b)x – ab

Thus, we have

(x + a)(x + b) = x\(^{2}\) + (b + a)x + ab

(x - a)(x - b) = x\(^{2}\) - (b + a)x + ab

(x + a)(x - b) = x\(^{2}\) + (a - b)x - ab

(x - a)(x + b) = x\(^{2}\) - (a - b)x – ab

(x + a)(x + b) = x\(^{2}\) + (Sum of constant terms)x + Product of constant terms.

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

1. Find the product of (z + 1)(z + 3) using the standard formula.

Solution:

We know, (x + a)(x + b) = x\(^{2}\) + (a + b)x + ab.

Therefore, (z + 1)(z + 3) = z\(^{2}\) + (1 + 3)z + 1 ∙ 3.

                                     = z\(^{2}\) + 4z + 3

2. Find the product of (m - 3)(m - 5) using the standard formula.

Solution:

We know, (x + a)(x + b) = x\(^{2}\) + (a + b)x + ab.

Therefore, (m - 3)(m - 5) = m\(^{2}\) + (-3 - 5)m + (-3) ∙ (-5).

                                      = m\(^{2}\) – 8m + 15

 

3. Find the product of (2a - 5)(2a + 3) using the standard formula.

Solution:

We know, (x + a)(x + b) = x\(^{2}\) + (a + b)x + ab.

Therefore, (2a - 5)(2a + 3) = (2a)\(^{2}\) + (-5 + 3) ∙ (2a) + (-5) ∙ 3.

                                        = 4a\(^{2}\) – 4a – 15.

4. Find the product: (2m + n – 3)(2m + n + 2).

Solution:

Product = {(2m + n) – 3}{(2m + n) + 2}

Let 2m + n = x. Then,

Product = (x – 3)(x + 2)

            = x\(^{2}\) + (-3 + 2)x + (-3) ∙ 2.

            = x\(^{2}\) – x – 6

Now plug-in x = 2m + n

                     = (2m + n)\(^{2}\)  - (2m + n) – 6

                     = (2m)\(^{2}\) + 2(2m)n + n\(^{2}\) – 2m – n – 6

                     = 4m\(^{2}\) + 4mn + n\(^{2}\) – 2m – n – 6

9th Grade Math

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