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Solve the given equation by the method of completing the square.
$9 x 2 − 15 x + 6 = 0$

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\frac{8}{3}

\frac{4}{3}

, 2

\frac{5}{3}

, 4

\frac{2}{3}

, 1

answer is D.

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Detailed Solution

The given quadratic equation is:

9 x 2 − 15 x + 6 = 0

.
We use the completing-the-square method.
On comparing given equation with

a x^{2} + bx + c = 0

we get a=9.
As

a = 9

, dividing the term by 9 to make the coefficient of

x 2

unity.

⇒ x 2 − 159 x + 69 = 0

⇒ x 2 − 53 x + 23 = 0

⇒ x 2 − 53 x = − 23 … … 1

Finding the value of

{(\frac{1}{2} coefficient of x)}^{2}

⇒ 12 × − 5 3 2 = − 5 6 2

Adding

{(\frac{- 5}{6})}^{2}

to both sides of equation (1),

⇒ x 2 − 53 x + − 5 6 2 = − 2 3 + − 5 6 2 ⇒ x 2 − 53 x + − 5 6 2 = − 2 3 + 2536

⇒ x − 56 2 = − 2 × 12 + 25 36 ⇒ x − 56 2 = 136

Taking square root on both sides of the obtained equation.

x − 56 = ± 136

⇒ x − 56 = ± 16

⇒ x = 5 ± 1 6

Simplifying to find the roots of the equation,

⇒ x = 5 + 1 6 = 1

And,

x = 5 − 1 6

= 23

Therefore, the roots of the equation are

x = 1 and x = \frac{2}{3} .

Hence, option 4 is correct.

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