Question

Solve the following quadratic equation by completing the square method, \[{x^2} + 10x + 24 = 0\].

Answer 1

Hint – Add and subtract something to the given equation to make a complete square.

Given equation ,
\[{x^2} + 10x + 24 = 0\]
When we have to solve a quadratic equation in complete square form then we add half of the coefficient of $x$ both sides .
That is we have to add \[{\left( {\dfrac{{10}}{2}} \right)^2}\] to both sides of the equation .
So we know,
\[{\text{ }}{\left( {\dfrac{{10}}{2}} \right)^2} = 25\]
After adding we get,
\[
  {x^2} + 10x + 24 + 25 = 25 \\
  {x^2} + 10x + 25 = 25 - 24 \\
\]
We can write it as,
\[{x^2} + 10x + 25 = {(x + 5)^2} = 1\]
On taking under root both sides we get,
\[
  x + 5 = \pm 1 \\
  x = - 5 + 1\,\,\,\,\,\& \,\,\,\,\,x = - 5 - 1 \\
\]
Then ,
\[x = - 6, - 4\]
So the value of $x = - 6, - 4$
Note – In these types of problems, we have to know the method of completing square .That is dividing all terms by a(the coefficient of ${x^2}$) . Then add the square of half of the coefficients of $x$ and then solve as above.