# Solve the following quadratic equation by completing the square method \[{{x}^{2}}+11x+30=0\].

Last updated date: 29th Mar 2023

•

Total views: 306.6k

•

Views today: 5.85k

Answer

Verified

306.6k+ views

Hint: Try to write the 11x in the form of 2ab. Rearrange them in LHS and RHS in order to get two perfect squares on both sides. Apply general algebra rules and extract the roots.

Complete step-by-step answer:

Given equation is \[{{x}^{2}}+11x+30=0\].

Or, we can write \[{{x}^{2}}+2(\dfrac{11}{2})x+30=0\].

\[\Rightarrow \] \[{{x}^{2}}+2(\dfrac{11}{2})x+{{(\dfrac{11}{2})}^{2}}+30-{{(\dfrac{11}{2})}^{2}}=0\] [Adding \[{{\left( \dfrac{11}{2} \right)}^{2}}\] on both sides]

Now, we know that,\[{{a}^{2}}+2ab+{{b}^{2}}={{(a+b)}^{2}}\]. Therefore, we can write the LHS in this form.

So, \[{{(x+\dfrac{11}{2})}^{2}}=\dfrac{121}{4}-30=\dfrac{121-120}{4}=\dfrac{1}{4}\]

Now, taking square roots on both sides, we get two equations as taking square roots gives both positive and negative values.

\[x+\dfrac{11}{2}=\dfrac{1}{2}\] ………. (1) and \[x+\dfrac{11}{2}=-\dfrac{1}{2}\]………… (2)

From (1) we get \[x=\dfrac{1}{2}-\dfrac{11}{2}=\dfrac{1-11}{2}=-\dfrac{10}{2}=-5\] and from (2) we get \[x=-\dfrac{1}{2}-\dfrac{11}{2}=-(\dfrac{1+11}{2})=-\dfrac{12}{2}=-6\].

These are the two roots of the given quadratic equation, which can be verified by putting the values of the roots in place of x and satisfying the equation.

Therefore, the roots of the given equation are -5 and -6 respectively.

Note: A quadratic equation must have at most (real/imaginary) two roots (all not necessarily distinct). Therefore, while taking square roots we should consider both positive and negative values. We can derive the well-known quadratic law for finding roots from this completing the square method. This method is the most fundamental method.

Complete step-by-step answer:

Given equation is \[{{x}^{2}}+11x+30=0\].

Or, we can write \[{{x}^{2}}+2(\dfrac{11}{2})x+30=0\].

\[\Rightarrow \] \[{{x}^{2}}+2(\dfrac{11}{2})x+{{(\dfrac{11}{2})}^{2}}+30-{{(\dfrac{11}{2})}^{2}}=0\] [Adding \[{{\left( \dfrac{11}{2} \right)}^{2}}\] on both sides]

Now, we know that,\[{{a}^{2}}+2ab+{{b}^{2}}={{(a+b)}^{2}}\]. Therefore, we can write the LHS in this form.

So, \[{{(x+\dfrac{11}{2})}^{2}}=\dfrac{121}{4}-30=\dfrac{121-120}{4}=\dfrac{1}{4}\]

Now, taking square roots on both sides, we get two equations as taking square roots gives both positive and negative values.

\[x+\dfrac{11}{2}=\dfrac{1}{2}\] ………. (1) and \[x+\dfrac{11}{2}=-\dfrac{1}{2}\]………… (2)

From (1) we get \[x=\dfrac{1}{2}-\dfrac{11}{2}=\dfrac{1-11}{2}=-\dfrac{10}{2}=-5\] and from (2) we get \[x=-\dfrac{1}{2}-\dfrac{11}{2}=-(\dfrac{1+11}{2})=-\dfrac{12}{2}=-6\].

These are the two roots of the given quadratic equation, which can be verified by putting the values of the roots in place of x and satisfying the equation.

Therefore, the roots of the given equation are -5 and -6 respectively.

Note: A quadratic equation must have at most (real/imaginary) two roots (all not necessarily distinct). Therefore, while taking square roots we should consider both positive and negative values. We can derive the well-known quadratic law for finding roots from this completing the square method. This method is the most fundamental method.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India