Question

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

Answer 1

Hint: This is a problem related to algebraic identities. This expression given above is of the form \[{a^2} - {b^2}\] such that 49 is the perfect square of 7. Thus on expanding this identity we will get the factors also. The factors are nothing but the values of x that satisfy the above expression. So we will first expand it and then find the answer.

Complete step-by-step answer:
Given that,
  \[{x^2} - 49\]
This can be written in the form \[{a^2} - {b^2}\]
  \[ \Rightarrow {x^2} - {7^2}\]
Now we know that, \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] so we can writs above expression as
  \[ \Rightarrow {x^2} - {7^2} = \left( {x + 7} \right)\left( {x - 7} \right)\]
Thus equating this to zero we get,
  \[ \Rightarrow \left( {x + 7} \right)\left( {x - 7} \right) = 0\]
Thus \[ \Rightarrow \left( {x + 7} \right) = 0\] or \[\left( {x - 7} \right) = 0\]
So value of x or factors of given expression are,
  \[ \Rightarrow x = - 7\] or \[x = 7\]
Thus the factors are \[x = \pm 7\] .
This is our final answer.
So, the correct answer is “ \[x = \pm 7\] ”.

Note: Factoring the expression is nothing but finding those values or values of variables that satisfy the given equation. If the equation is having only one degree that the power of variable is 1 then that expression has only one value fixed that satisfies the equation. Whereas if the equation has degree 2 then there are two values of that variable that satisfies the expression.
Like above putting \[7\] in the expression we get, \[ \Rightarrow {x^2} - {7^2} \Rightarrow {7^2} - {7^2} = 0\]
And then putting \[ - 7\] we get, \[ \Rightarrow {x^2} - {7^2} \Rightarrow {\left( { - 7} \right)^2} - {7^2} = 0\]
The number of factors of the given expression is equal to the degree of that expression.
Equation with degree 3 is called cubic equation.