Factorise $x^{2} + 10 x + 8$ by adding and subtracting appropriate quantity

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Solution

In the given equation $x^{2} + 10 x + 8$ , add and subtract $5^{2}$ to make it a perfect square as shown below:

$x^{2} + 10 x + 8 = (x^{2} + 10 x + 5^{2}) + 8 - 5^{2} = (x^{2} + (2) (5) x + 5^{2}) + 8 - 25 = (x + 5)^{2} - 17$

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

Using the above identity, the expression $(x + 5)^{2} - 17$ can be factorised as follows:

$(x + 5)^{2} - 17 = (x + 5)^{2} - (\sqrt{17})^{2} = (x + 5 - \sqrt{17}) (x + 5 + \sqrt{17})$

Hence, $x^{2} + 10 x + 8 = (x + 5 - \sqrt{17}) (x + 5 + \sqrt{17})$

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