Question

How do you factor $ \left( {{x^2} - 5} \right) $ ?

Answer 1

Hint: In this question, we need to factorize $ \left( {{x^2} - 5} \right) $ . We will use the difference of square identity here. We will convert the factor similar to the identity, by using the condition square root. Then expand it in accordance with the formula of the identity which is the required factor.

Complete step-by-step answer:
Now, we need to factorize $ \left( {{x^2} - 5} \right) $ .
We know that, the difference of square identity,
 $ {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) $
To apply this, we need to bring $ \left( {{x^2} - 5} \right) $ into the form of $ {a^2} - {b^2} $ .
$\Rightarrow \sqrt 5 \times \sqrt 5 = \sqrt {25} = 5 $
It is known as squaring a square root. Squaring a number and taking the square root of a number are opposite operations; thus, they undo each other. The result of squaring a square root, then, is simply the number under the radical sign.
Therefore, we can write $ \left( {{x^2} - 5} \right) $ as $ {x^2} - {\left( {\sqrt 5 } \right)^2} $ , which is in the form of $ {a^2} - {b^2} $ .
Then, $ {x^2} - {\left( {\sqrt 5 } \right)^2} = \left( {x + \sqrt 5 } \right)\left( {x - \sqrt 5 } \right) $
Hence, the factors are $ \left( {x + \sqrt 5 } \right) $ and $ \left( {x - \sqrt 5 } \right) $ .
So, the correct answer is “ $ \left( {x + \sqrt 5 } \right) $ and $ \left( {x - \sqrt 5 } \right) $ ”.

Note: In this question it is important to note that a radicand is a number underneath the radical sign. To multiply radicands, multiply the numbers as if they were whole numbers. Make sure to keep the product under one radical sign. Then factor out any perfect squares in the radicand. If it is not a perfect square, then place the square root of the perfect square in front of the radical sign. When the equation is given in the form of a quadratic equation, $ a{x^2} + bx + c $ then we need to find two numbers that multiply to give $ ac $ , and add to give $ b $ which is called sum-product pattern. Then rewrite the middle with those numbers. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor.