Answer

Verified

390.3k+ views

**Hint:**Here, we will simply add and subtract square of a constant and write the term containing $n$ such that it forms a term in a form $2ab$, thus, this will help us to apply the square identities and thus, ‘complete the square’ and solve it further to find the required roots of the given quadratic equation.

**Formula Used:**

${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

**Complete step-by-step answer:**

The given quadratic equation is:

$2{n^2} - 7n + 3 = 0$

Now, dividing both sides by 2, we get,

${n^2} - \dfrac{7}{2}n + \dfrac{3}{2} = 0$

Now, since, we are required to solve this question using completing the square, hence, we will write this quadratic equation as:

${\left( n \right)^2} - 2\left( n \right)\left( {\dfrac{7}{4}} \right) + {\left( {\dfrac{7}{4}} \right)^2} - {\left( {\dfrac{7}{4}} \right)^2} + \dfrac{3}{2} = 0$

Hence, even if this quadratic equation was not a perfect square, but we tried to make it a perfect square by adding and subtracting the square of a constant such that, we complete the square by making the identity, ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$

Hence, using this identity, we get,

${\left( {n - \dfrac{7}{4}} \right)^2} - \dfrac{{49}}{{16}} + \dfrac{3}{2} = 0$

$ \Rightarrow {\left( {n - \dfrac{7}{4}} \right)^2} + \dfrac{{ - 49 + 24}}{{16}} = 0$

Hence, we get,

$ \Rightarrow {\left( {n - \dfrac{7}{4}} \right)^2} - \dfrac{{25}}{{16}} = 0$

Adding $\dfrac{{25}}{{16}}$ on both sides,

$ \Rightarrow {\left( {n - \dfrac{7}{4}} \right)^2} = \dfrac{{25}}{{16}}$

$ \Rightarrow {\left( {n - \dfrac{7}{4}} \right)^2} = {\left( {\dfrac{5}{4}} \right)^2}$

Taking square root on both sides, we get

$ \Rightarrow \left( {n - \dfrac{7}{4}} \right) = \pm \dfrac{5}{4}$

Adding $\dfrac{7}{4}$ on both the sides, we get,

$ \Rightarrow n = \pm \dfrac{5}{4} + \dfrac{7}{4}$

Hence,

$n = \dfrac{5}{4} + \dfrac{7}{4} = \dfrac{{5 + 7}}{4} = \dfrac{{12}}{4} = 3$

Or $n = - \dfrac{5}{4} + \dfrac{7}{4} = \dfrac{{7 - 5}}{4} = \dfrac{2}{4} = \dfrac{1}{2}$

Therefore, the roots of the given quadratic equation $2{n^2} - 7n + 3 = 0$ are 3 and $\dfrac{1}{2}$

Thus, this is the required answer.

**Note:**

If in the question, it was not mentioned that we have to use the method of completing the square, then, we could have used the quadratic formula to solve the given quadratic equation.

Given quadratic equation is $2{n^2} - 7n + 3 = 0$

Now, dividing both sides by 2, we get,

${n^2} - \dfrac{7}{2}n + \dfrac{3}{2} = 0$

Comparing this with the general quadratic equation i.e. $a{x^2} + bx + c = 0$

We have,

$a = 1$, $b = \dfrac{{ - 7}}{2}$ and $c = \dfrac{3}{2}$

Now, we can find the roots of a quadratic equation using the quadratic formula, $n = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Hence, for the equation ${n^2} - \dfrac{7}{2}n + \dfrac{3}{2} = 0$, substituting$a = 1$, $b = \dfrac{{ - 7}}{2}$and $c = \dfrac{3}{2}$, we get

$n = \dfrac{{\dfrac{7}{2} \pm \sqrt {{{\left( {\dfrac{{ - 7}}{2}} \right)}^2} - 4\left( 1 \right)\left( {\dfrac{3}{2}} \right)} }}{{2\left( 1 \right)}}$

$ \Rightarrow n = \dfrac{{\dfrac{7}{2} \pm \sqrt {\dfrac{{49}}{4} - 6} }}{2} = \dfrac{{\dfrac{7}{2} \pm \sqrt {\dfrac{{49 - 24}}{4}} }}{2} = \dfrac{{\dfrac{7}{2} \pm \sqrt {\dfrac{{25}}{4}} }}{2}$

Solving further, we get,

\[ \Rightarrow n = \dfrac{{\dfrac{7}{2} \pm \dfrac{5}{2}}}{2} = \dfrac{{7 \pm 5}}{4}\]

Hence, we get,

\[n = \dfrac{{7 + 5}}{4} = \dfrac{{12}}{4} = 3\]

Or \[n = \dfrac{{7 - 5}}{4} = \dfrac{2}{4} = \dfrac{1}{2}\]

Therefore, the roots of the given quadratic equation $2{n^2} - 7n + 3 = 0$ are 3 and $\dfrac{1}{2}$

Thus, this is the required answer.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

One cusec is equal to how many liters class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

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

What organs are located on the left side of your body class 11 biology CBSE