Question

Find the roots of the equation $5{{x}^{2}}-6x-2=0$ by the method of completing the square.
(a)$\dfrac{3\pm \sqrt{76}}{2}$
(b)$\dfrac{3\pm \sqrt{19}}{4}$
(c)$\dfrac{3\pm \sqrt{76}}{10}$
(d)$\dfrac{3\pm \sqrt{19}}{5}$

Answer 1

Hint: Completing the Square is a method that may be used for any quadratic equation. By adjusting your constant, you can create a perfect square on the left side of the equation. A perfect square can be factored into two identical binomials, which you can use to solve for any valid values of x.

Complete step-by-step answer:
Suppose $a{{x}^{2}}+bx+c=0$ is the given quadratic equation. Then follow the given steps to solve it by completing the square method.

Write the equation in the form, such that c is on the right side.

If a is not equal to 1, then divide the complete equation by a, such that the coefficient of ${{x}^{2}}$ is 1.

Now add the square of half of the coefficient of term-x, ${{\left( \dfrac{b}{2a} \right)}^{2}}$on both sides.

Factorize the left side of the equation as the square of the binomial term.

Take the square root on both the sides

Solve for variable x and find the roots.

Let us consider the quadratic equation,

$5{{x}^{2}}-6x-2=0$

Dividing both sides by 5, we get

\[{{x}^{2}}-\left( \dfrac{6}{5} \right)x-\left( \dfrac{2}{5} \right)=0\]

Add and subtract the term ${{\left( \dfrac{3}{5} \right)}^{2}}$ on the left side of the equation, we get

\[{{x}^{2}}-\left( \dfrac{6}{5} \right)x+{{\left( \dfrac{3}{5} \right)}^{2}}-{{\left( \dfrac{3}{5} \right)}^{2}}-\left( \dfrac{2}{5} \right)=0\]

Rearranging the terms, we get

\[{{x}^{2}}-2\left( \dfrac{3}{5} \right)x+{{\left( \dfrac{3}{5} \right)}^{2}}=\dfrac{9}{25}+\dfrac{2}{5}\]

\[{{\left( x-\dfrac{3}{5} \right)}^{2}}=\dfrac{19}{25}\]

Squaring roots on the both sides, we get

\[\left( x-\dfrac{3}{5} \right)=\pm \sqrt{\dfrac{19}{25}}\]

\[x=\dfrac{3}{5}\pm \dfrac{\sqrt{19}}{5}\]

$x=\dfrac{3\pm \sqrt{19}}{5}$

Therefore, the correct option is (d).

Note: Completing the square method is one of the methods to find the roots of the given quadratic equation. A polynomial equation with degree equal to two is known as a quadratic equation. Quad means four but Quadratic means to make square.