Factorise $x^{2} + 10 x + 20$ by adding and subtracting appropriate quantity

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Solution

In the given equation $x^{2} + 10 x + 20$ , add and subtract $5^{2}$ to make it a perfect square as shown below:

$x^{2} + 10 x + 20 = (x^{2} + 10 x + 5^{2}) + 20 - 5^{2} = (x^{2} + (2) (5) x + 5^{2}) + 20 - 25 = (x + 5)^{2} - 5$

Since we know the identity $a^{2} - b^{2} = (a + b) (a - b)$

Using the above identity, the equation $(x + 5)^{2} - 5$ can be factorised as follows:

$(x + 5)^{2} - 5 = (x + 5)^{2} - (\sqrt{5})^{2} = (x + 5 - \sqrt{5}) (x + 5 + \sqrt{5})$

Hence, $x^{2} + 10 x + 20 = (x + 5 - \sqrt{5}) (x + 5 + \sqrt{5})$

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