Question

How do you solve by completing the square ${x^2} + 2x - 5 = 0?$

Answer 1

Hint: As we know that above given expression is a quadratic polynomial or quadratic equation. An equation that can be written in the form of $a{x^2} + bx + c$is called a quadratic polynomial, where the term $x$ represents the unknown and $a,b$and $c$represents the known numbers. Completing the square means that we have to turn the perfect square trinomial or binomial on the left side of the equation and then we solve for $x$.

Complete step-by-step solution:
As per the given question we have an equation ${x^2} + 2x - 5 = 0$, we have to solve it by completing the square. We will first add $5$ to the both sides of the equation: ${x^2} + 2x - 5 + 5 = 0 + 5 \Rightarrow {x^2} + 2x = 5$. The coefficient of $x$ is $2$ so we will divide the coefficient of $x$ by $2$, then we will square the result, i.e. $\dfrac{2}{2} = 1$, now we will square it ${1^2} = 1$. So the value is $1$,
We will add the value to the both sides of the equation: ${x^2} + 2x + 1 = 5 + 1 \Rightarrow {x^2} + 2x + 1 = 6$.
We can write the left side ${x^2} + 2x + 1 = {(x + 1)^2}$. It is a perfect square trinomial. So by substituting the value we get: ${(x + 1)^2} = 6 \Rightarrow x + 1 = \pm \sqrt 6 $. So the value of $x$ is $\sqrt 6 - 1$ or $( - \sqrt 6 - 1)$.

Hence the required answer is $ \pm \sqrt 6 - 1$.

Note: In this type of question where it is mentioned that we have to use the square method to solve, we have to memorize the steps involved in the method. Whenever we take the square root of any quantity then we should take both the positive and the negative values as it gives both. Also we should write both the values of x.