How do you solve using the completing the square method 9(x^2) - 18x = -3?

Apr 10, 2018

You need to use the formula: ${\left(a - b\right)}^{2} = {a}^{2} - 2 a b + {b}^{2}$

Explanation:

If we manipulate the equation given to get $9 {x}^{2} - 18 x + 3 = 0$, using the formula above we can see that:

$9 {x}^{2} = {a}^{2}$, and that $- 18 x = - 2 a b$

So from the first we have $3 x = a$ and substituting this in the second we have

$- 6 \cdot 3 x = - 6 a = - 2 a b$, and then

$- 6 = - 2 b$ and from here $b = 3$

Now completing the square in the equation given :

$9 {x}^{2} - 18 x + 3 = 9 {x}^{2} - 18 x + 9 - 6 = 0 = {\left(3 x - 3\right)}^{2} - 6 = 0$

that is:

${\left(3 x - 3\right)}^{2} = 6$, and we have two solutions:

$\left(3 x - 3\right) = + \sqrt{6}$
$\left(3 x - 3\right) = - \sqrt{6}$, and then:
$x = \frac{3 + \sqrt{6}}{3}$ and
$x = \frac{3 - \sqrt{6}}{3}$