Question

How do you factor the expression \[{x^2} - 49\]?

Answer 1

Hint: Factoring reduces the higher degree equation into its linear equation. The above given question has the highest power 2 and it can be reduced by using the formula for the difference of the two squares but for that both the terms should be the perfect square of a number.

Complete step by step solution:
Given, the expression has terms with the highest power of 2. We can use the formula of difference of two squares. The difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. The formula for difference of square of two terms is:
\[{a^2} - {b^2} = \left( {a + b} \right)(a - b)\]
First, we need to group the first term as a whole square because each term needs to be a perfect square.
Perfect square means the term a should be the square of another number. For example,9 is the square of the number 3. Hence 9 will be a perfect square.
Then by writing the above terms in perfect square we get,
\[{\left( x \right)^2} - {(7)^2}\]
Since 49 is a perfect square of 7.
So, by further solving we get,
\[{\left( x \right)^2} - {\left( 7 \right)^2} = \left( {x + 7} \right)\left( {x - 7} \right)\]
Therefore, by solving the above expression we get the above solution for the expression.

Note: An important thing to note is that if the first term becomes negative then using the above formula is not valid. For example if \[{x^2}\]changes to \[ - {x^2}\] in the expression \[ - {x^2} - {\left( 7 \right)^2}\]then the meaning of the expression completely changes so in this case we cannot use the above given formula.