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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)
= x2 + xb + ax + ab
= x2 + (b + a)x + ab
(x - a)(x - b) = x(x - b) - a (x - b)
= x2 - xb - ax + ab
= x2 - (b + a)x + ab
(x + a)(x - b) = x(x - b) + a (x - b)
= x2 - xb + ax - ab
= x2 + (a - b)x - ab
(x - a)(x + b) = x(x + b) - a (x + b)
= x2 + xb - ax - ab
= x2 - (a - b)x – ab
Thus, we have
(x + a)(x + b) = x2 + (b + a)x + ab
(x - a)(x - b) = x2 - (b + a)x + ab
(x + a)(x - b) = x2 + (a - b)x - ab
(x - a)(x + b) = x2 - (a - b)x – ab
(x + a)(x + b) = x2 + (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) = x2 + (a + b)x + ab.
Therefore, (z + 1)(z + 3) = z2 + (1 + 3)z + 1 ∙ 3.
= z2 + 4z + 3
2. Find the product of (m - 3)(m - 5) using the standard formula.
Solution:
We know, (x + a)(x + b) = x2 + (a + b)x + ab.
Therefore, (m - 3)(m - 5) = m2 + (-3 - 5)m + (-3) ∙ (-5).
= m2 – 8m + 15
3. Find the product of (2a - 5)(2a + 3) using the standard formula.
Solution:
We know, (x + a)(x + b) = x2 + (a + b)x + ab.
Therefore, (2a - 5)(2a + 3) = (2a)2 + (-5 + 3) ∙ (2a) + (-5) ∙ 3.
= 4a2 – 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)
= x2 + (-3 + 2)x + (-3) ∙ 2.
= x2 – x – 6
Now plug-in x = 2m + n
= (2m + n)2 - (2m + n) – 6
= (2m)2 + 2(2m)n + n2 – 2m – n – 6
= 4m2 + 4mn + n2 – 2m – n – 6
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