Question

How do you solve ${x^2} + x - 6 = 0$ by completing the square?

Answer 1

Hint: Completing the square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial.
To solve $a{x^2} + bx + c = 0$ by completing the square:
Transform the equation so that the constant term, $c$, is alone on the right side.
If $a$, the leading coefficient (the coefficient of the ${x^2}$ term) is not equal to $1$, divide both sides by $a$
Add the square of half the coefficient of the x-term, ${\left( {\dfrac{b}{{2a}}} \right)^2}$to both sides of the equation.
Factor the left side as the square of a binomial.
Take the square root of both sides.

Complete step-by-step solution:
Firstly, move the $ - 6$ to the right side-
$\therefore ,{x^2} + x = 6$
Now,
Complete the square by adding ${\left( {\dfrac{b}{2}} \right)^2}$to both sides, where $b$is the second coefficient (in this case, $b = 1$):
$
\Rightarrow {x^2} + x + {\left( {\dfrac{1}{2}} \right)^2} = 6 + {\left( {\dfrac{1}{2}} \right)^2} \\
\Rightarrow {x^2} + x + \dfrac{1}{4} = 6 + \dfrac{1}{4} \\
\Rightarrow {\left( {x + \dfrac{1}{2}} \right)^2} = \dfrac{{25}}{4} \\
 $
Simplifying:
$
\Rightarrow x + \dfrac{1}{2} = \pm \dfrac{5}{2} \\
\Rightarrow x = - \dfrac{1}{2} \pm \dfrac{5}{2} \\
\Rightarrow x = \dfrac{4}{2} \, or - \dfrac{6}{2} \\
\Rightarrow x = 2 \, or - 3 \\
 $

Additional Information:
What is completing by square?
When you have a polynomial such as
${x^2} + 4x = 20$
It is sometimes desirable to express it in the form of
${a^2} + {b^2}$
To do this, we can artificially introduce a constant which allows us to factor a perfect square out of the expression like so:
$
\Rightarrow {x^2} + 4x + 20 \\
 \Rightarrow {x^2} + 4x + 4 - 4 + 20 \\
 $
Notice that by simultaneously adding and subtracting, we have not changed the value of expression.
Now we can do this
$
  \Rightarrow \left( {{x^2} + 4x + 4} \right) + \left( {20 - 4} \right) \\
  \Rightarrow {\left( {x + 2} \right)^2} + 16 \\
  \Rightarrow {\left( {x + 2} \right)^2} + {4^2} \\
 $

Note: The roots of polynomials are the values of $x$which satisfy the equation. There are several methods to find the roots of a quadratic equation. One of them is completing the squares which is shown in the above given question. This is one of the most accurate methods to solve the quadratic equations.