Question

How do you solve an equation by completing the square?

Answer 1

Hint:In the given question, we have been asked to write the method to complete the square. This is done by taking the whole equation, seeing which element is missing, adding and subtracting that element, and then condensing it into the formula of the whole square.

Complete step by step answer:
We can solve an equation in many ways – using determinants, cross-multiplication, et cetera. Completing the square is another method to solve an equation.
In the following steps, we are going to learn how we can solve an equation by completing the square.
We have a quadratic equation \[a{x^2} + bx + c = 0\] (where \[a,b,c\] are constants) which is to be solved by completing the square.
First, we divide both the sides by \[a\], we get,
\[{x^2} + \dfrac{b}{a}x + \dfrac{c}{a} = 0\]
Let \[\dfrac{b}{a} = m\] and \[\dfrac{c}{a} = n\]
So, we get,
\[{x^2} + mx + n = 0\]
Now, we shift the constant \[n\] to the other side, we get,
$\Rightarrow$ \[{x^2} + mx = - n\]
Now, add the square of half of \[m\] or \[{\left( {\dfrac{m}{2}} \right)^2}\] to both sides,
$\Rightarrow$ \[{x^2} + 2 \times \dfrac{1}{2}m \times x + {\left( {\dfrac{m}{2}} \right)^2} = {\left( {\dfrac{m}{2}} \right)^2} - n\]
Now, we factorize the left-hand side into a perfect square using the whole square sum formula,
$\Rightarrow$ \[{\left( {x + \dfrac{m}{2}} \right)^2} = \dfrac{{{m^2} - 4n}}{4}\]
Now, we take square root on both sides,
$\Rightarrow$ \[x + \dfrac{m}{2} = \pm \dfrac{{\sqrt {{m^2} - 4n} }}{2}\]
Taking \[\dfrac{m}{2}\] to the other side,
$\Rightarrow$ \[x = \pm \dfrac{{\sqrt {{m^2} - 4n} }}{2} - \dfrac{m}{2}\]
Substituting back the values of \[b\] and \[c\],
$\Rightarrow$ \[x = \pm \dfrac{{\sqrt {{{\left( {b/a} \right)}^2} - 4\left( {c/a} \right)} }}{2} - \dfrac{b}{{2a}}\]
Then we just solve for \[x\] and find the answer.

Note: In this question, we learned how we can solve an equation by completing its square. We need to know the exact steps lest we are going to get the wrong answer. So, we need to pay attention every step of the way so that we get the correct answer.