Question

How do you factor and solve ${x^2} = 81$?

Answer 1

Hint: Here we can write the above term as ${x^2} - 81 = 0$ and then solve by using the formula of ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ as we know that $81 = {9^2}$ and this way we can also find the factors of the above given equation.

Complete step by step solution:
Here we are given to find the factors of ${x^2} = 81$
So we know that when we are given the term in the form of $a = b$ we can write it as $a - b = 0$ also.
Similarly here we are given ${x^2} = 81$ and therefore we can write it as ${x^2} - 81 = 0$$ - - - - (1)$
Now we know that we can express the term $81$ in the form of a square as we know that when $9$ is multiplied by itself it gives us the result as $81$.
Hence we can write that $81 = {9^2}$
Now substituting this value in the equation (1) we will get:
${x^2} - {9^2} = 0$
Now we know that this equation has been converted in the form of ${a^2} - {b^2}$.
Hence we can apply the formula here of ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
We will get:
${x^2} - {9^2} = \left( {x + 9} \right)\left( {x - 9} \right)$
This is the simplification of the above given term. Now we can find its factors by equating it to zero and we will get:
$
  {x^2} - {9^2} = \left( {x + 9} \right)\left( {x - 9} \right) = 0 \\
  x = 9{\text{ or }} - 9 \\
 $
Hence this is the method we can apply to get the factors and solve the equation of the form ${a^2} - {b^2}$.

Note:
Here the student must know that if he is given the equation of the form ${a^3} - {b^3}$ then we can apply the formula of ${a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)$.