Topic Content:
- Factorizing the Difference of Two Perfect Squares
A theorem that tells us if a quadratic equation can be written as a product of two binomials, in which one shows the difference of the square roots and the other shows the sum of the square roots, is called the difference of two squares.
Expressions like x2 – 4, p2 – 81 and y2 – 1 are called the difference of two perfect squares.
⇒ \( \scriptsize x^2 \: – \: 4 \\ = \scriptsize x^2 \: – \: 2^2 \)
⇒ \( \scriptsize p^2 \: – \: 81 \\ = \scriptsize p^2 \: – \: 9^2 \)
⇒ \( \scriptsize y^2 \: – \: 1 \\ = \scriptsize y^2 \: – \: 1^2 \)
Other examples of perfect squares are 1, 4, 9, 16, 25, 36, ….
Also even powers of x such as x2, x4, x6, x8,…… are perfect squares.
Recall:
= \( \scriptsize (x \: – \: 1)(x\: +\: y) = x^2 \: +\: xy \; – \: xy \: +\: y^2\)
= \( \scriptsize x^2 \: -\: y^2 \)
⇒ \( \scriptsize x^2 \: -\: y^2 = (x \: -\: 1)(x\: + \: y) \)
Example 6.3.1:
Factorise the following:
(a) \( \scriptsize 25 x^2 – 16y^2\)
(b) \( \scriptsize 2x^2 – 8(x + y)^2 \)
(c) \( \scriptsize 3 \frac{6}{25} x^2 \; – \; \frac{9}{16} y^2 \)
(d) \( \scriptsize \sqrt {81 x^4 } \; – \; \normalsize \frac{9}{4} \)
Solution
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