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

Last updated date: 27th Mar 2023

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Answer

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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.

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.

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