Is is possible to factorise $x^{2} + 4 x + 20$ ? Give reasons.

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Solution

In the given equation $x^{2} + 4 x + 20$ , add and subtract $2^{2}$ to make it a perfect square as shown below:

$x^{2} + 4 x + 20 = (x^{2} + 4 x + 2^{2}) + 20 - 2^{2} = (x^{2} + (2) (2) x + 2^{2}) + 20 - 4 = (x + 2)^{2} + 16 = (x + 2)^{2} + (4)^{2}$

We cannot factorise it further because it is in the form $a^{2} + b^{2}$

Hence, $x^{2} + 4 x + 20$ cannot be factorised.

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The interval that gives all possible values of the expression $\sqrt{x^{2} - 4 x + 20}$ is

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