Question

Solve the given quadratic equation for n : \[4{{n}^{2}}+5n-636=0\]

Answer 1

- Hint: To solve this type of problem first we have to find the value of discriminant and say how the roots will be. Then we have to write the values in the formula to get the roots. Nature of the roots is given by \[{{b}^{2}}-4ac\]. By substituting the values of a, b, c in this formula gives us the roots. \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].

Complete step-by-step solution -

Given equation is \[4{{n}^{2}}+5n-636=0\] .
The value of a, b, c is noted.
\[a=4,b=5,c=-636\]
Discriminant: \[{{b}^{2}}-4ac\]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
Substituting the values in (1) we get,
\[{{\left( 5 \right)}^{2}}-4\left( 4 \right)\left( -636 \right)\]
\[D=25+10176\]
\[D=25+10176=10201\]
Therefore, the discriminant is greater than zero, which means the roots are real and unequal.
To find the roots we have the formula,

\[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Now substituting the values of a, b, c in (2) we get,
\[\dfrac{-5\pm \sqrt{{{\left( 5 \right)}^{2}}-4\left( 4 \right)\left( -636 \right)}}{2\left( 4 \right)}\]
\[=\dfrac{-5\pm \sqrt{25+10176}}{8}\]
\[=\dfrac{-5\pm \sqrt{10201}}{8}\]
Now we know that 10201 is the square of 101. So, we get
\[=\dfrac{-5\pm 101}{8}\]
Therefore, the roots of the equation \[4{{n}^{2}}+5n-636=0\] is:
\[\dfrac{-5+101}{8},\dfrac{-5-101}{8}\]
\[12,-\dfrac{53}{4}\]

The roots for the quadratic equation \[4{{n}^{2}}+5n-636=0\] are \[12,-\dfrac{53}{4}\] .

Note: In (2) the term under root is not zero. The discriminant plays a major role which specifies the nature of roots. The roots can be either real or imaginary. This is a direct problem with using mathematical operations. Be careful while doing calculations. You should also remember that in case of quadratic equations with rational coefficients, the irrational and complex roots appear in conjugate pairs.