Question

How do you solve ${x^2} - 1 = 0$ ?

Answer 1

Hint:The given problem requires us to solve a quadratic equation. There are various methods that can be employed to solve a quadratic equation like completing the square method, using quadratic formula and by splitting the middle term. We will be using some of the basic algebraic identities in solving the given problem such as ${a^2} - {b^2} = \left( {a - b} \right)\left( {a - b} \right)$.

Complete step by step answer:
In the given question, we are required to solve the equation ${x^2} - 1 = 0$.We can solve the given equation by any of the methods. Consider the equation ${x^2} - 1 = 0$.The equation can be factorized easily by using the algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a - b} \right)$. So, resembling the left side of the identity, we get,
$ \Rightarrow {\left( x \right)^2} - {\left( 1 \right)^2} = 0$

Now, using the identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a - b} \right)$.
$ \Rightarrow \left( {x - 1} \right)\left( {x + 1} \right) = 0$
Now, since the product of two factors is zero. So, either of the two factors must be zero. Hence, we get,
either $x - 1 = 0$ or $x + 1 = 0$.
Either $x = 1$ or $x = - 1$ .

So, the roots of the given equation ${x^2} - 1 = 0$ are: $x = 1$ and $x = - 1$.

Note:Quadratic equations are the polynomial equations with degree of the variable or unknown as $2$. Quadratic equations can be solved by splitting the middle term, factoring common factors, using the quadratic formula and completing the square method. But we should first look for simpler and the basic methods such as applications of algebraic identities.