Question

Find the solution of \[{x^2} + 10x + 24 = 0\] using completing the square.

Answer 1

Hint: A perfect square trinomial on the left side of the equation should be created. Then it should be factored and hence solve for “x”. The general equation for a perfect square trinomial is \[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\] . 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.

Complete step-by-step answer:
We can have the expansion of \[{\left( {x + 5} \right)^2}\] as \[{x^2} + 10x + 25\] by the formula \[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\] .
Hence,
 \[{\left( {x + 5} \right)^2} = {x^2} + 10x + 25\]
So we have,
 \[ \Rightarrow 0 = {x^2} + 10x + 24 = {\left( {x + 5} \right)^2} - 1\]
Since according to the question we have \[{x^2} + 10x + 24 = 0\] that is why to obtain this we can subtract \[1\] from \[{\left( {x + 5} \right)^2}\] .
Now adding 1 to both the ends to eliminate 1 from the equation and obtain \[{\left( {x + 5} \right)^2}\] we have,
 \[ \Rightarrow {\left( {x + 5} \right)^2} = 1\]
Removing the square and taking root to the left hand side we have,
 \[ \Rightarrow x + 5 = \pm \sqrt 1 = \pm 1\]
Now subtracting 5 from both the sides to have “x” in the right hand side we have,
 \[ \Rightarrow x = - 5 \pm 1\]
Solving the above equation for “x” gives the values \[x = - 6\] or \[x = - 4\] .
So, the correct answer is “ \[x = - 6\] or \[x = - 4\] .”.

Note: To solve an equation by completing the square method the constant term should be alone on the right side. If the leading coefficient “a” that is the coefficient of \[{x^2}\] terms not equal to 1 then both sides should be divided by “a”. The square of half the coefficient of the x-term should be added to both sides of the equation and the left side should be factored as the square of a binomial. Both the sides should be taken square root and then x should be solved.