Question

Solve the quadratic equation 9x^2 - 15x + 6 = 0 by method of Completing the square.

Answer 1

Given equation : 9x² - 15x + 6 = 0

$\therefore 3x^{2} - 5x + 2 = 0$



Solving the given equation according to the method of completing square.

Step 1 : Dividing the whole equation by the co efficient of x², in the question coefficient of x² is 3.
$\therfore \text{Dividing the whole equation by 3 }$



$\rightarrow \: \dfrac{3x^{2} - 5x + 2 }{3} = \dfrac{0}{3} \\ \\ \\ \rightarrow {x}^{2} - \dfrac{5x}{3} + \dfrac{2}{3} = 0 \\ \\ \\ \rightarrow {x}^{2} - \dfrac{5}{3} x = - \dfrac{2}{3}$

.

Step 2 : adding ( 5 / 6 )² on both sides.


$\rightarrow {x}^{2} - \dfrac{5x}{3} + \bigg( \dfrac{5}{6} \bigg) {}^{2} = - \dfrac{2}{3} + \bigg( \dfrac{5}{6} \bigg) {}^{2}$


a² - 2ab + b² = ( a - b )²
By Using the formula given above.
${x}^{2} - \dfrac{5x}{3} + \bigg( \dfrac{5}{6} \bigg) {}^{2} = \bigg( x - \dfrac{5}{3} \bigg)^{2}$


$\rightarrow \bigg( x - \dfrac{5}{6} \bigg)^{2} = \dfrac{ - 2}{3} + \dfrac{25}{36} \\ \\ \\ \rightarrow \bigg( x - \dfrac{5}{6} \bigg)^{2} = \dfrac{ - 24 + 25}{36} \\ \\ \\ \rightarrow \bigg( x - \dfrac{5}{6} \bigg)^{2} = \dfrac{1}{36} \\ \\ \\ \rightarrow \bigg( x - \dfrac{5}{6} \bigg) = \pm \dfrac{ 1}{6} \\ \\ \\ \rightarrow \: x = \frac{1}{6} + \dfrac{5}{6} \: \: or \: \: \dfrac{5}{6} - \dfrac{1}{6}$




$\rightarrow x = 1 or \dfrac{2}{3}$

Answer 2

Given Equation is 9x^2 - 15x + 6 = 0.

(i)

Divide throughout by 9.

⇒ x^2 - (15/9) x + (6/9) = 0

⇒ x^2 - (5/3) x + (2/3) = 0

(ii)

Rewrite the equation with the constant term on the right side.

⇒ x^2 - (5/3) x = -2/3

(iii)

Complete the square by adding the square of one-half of coefficient.

⇒ x^2 - (5/3)x  + (5/6)^2 = -2/3 + (5/6)^2

⇒ x^2 - (5/3) x + (5/6)^2 = 1/36

(iv)

Write the left side as square.

⇒ (x - 5/6)^2 = 1/36

⇒ x - 5/6 = √1/36

⇒ x - 5/6 = 1/6

(v)

Equate and solve.

(a)

⇒ x - 5/6 = 1/6

⇒ x = 1/6 + 5/6

⇒ x = 1.

(b)

⇒ x - 5/6 = -1/6

⇒ x = -1/6 + 5/6

⇒ x = 2/3.

Therefore, roots are 1 and 2/3.

Hope it helps!