Question

Solve for x.

Answer 1

Answer:

$x = \frac{2}{3} + \frac{\sqrt{10} }{3} , x = \frac{2}{3} - \frac{\sqrt{10} }{3}$

Step-by-step explanation:

Multiplying the coefficient of x by the constant to get:

3 x (-2) = -6

Find factors of -6 that equal the middle term -4.

Since, no such factors can be found so we can solve the equation by completing the square.

To complete the square, divide the equation by the coefficient of x^2 which is 3 to get:

$x^{2} - \frac{4}{3} x - \frac{2}{3} = 0$

$x^{2} - \frac{4}{3} x = \frac{2}{3}$

Now divide the coefficient of x by 2 and add the square of the result to both sides of the equation:

$x^{2} - \frac{4}{3} x + (-\frac{2}{3} )^{2} = \frac{2}{3} + (-\frac{2}{3} )^{2}$

$(x - \frac{2}{3} )^2 = \frac{2}{3} + \frac{4}{9}$

$(x - \frac{2}{3} )^2 = \frac{4}{9}$

$\sqrt{(x - \frac{2}{3} )^2} = \sqrt{\frac{4}{9} }$

$x - \frac{2}{3} = \sqrt{\frac{10}{9} }$ , $x - \frac{2}{3} = -\sqrt{\frac{10}{9} }$

$x = \frac{2}{3} + \frac{\sqrt{10} }{3} , x = \frac{2}{3} - \frac{\sqrt{10} }{3}$

Answer 2

ANSWER

$x=\frac{2-\sqrt{10}} {3}$

or

$x=\frac{\sqrt{10}+2} {3}$

We have

$3x^2-4x-2=0$

Since we cannot factor easily, we complete the square.

Adding 2 to both sides give,

$3x^2-4x=2$

Dividing through by 3 gives

$x^2-\frac{4}{3}x= \frac{2}{3}$

Adding $(-\frac{2}{3})^2$ to both sides gives

$x^2-\frac{4}{3}x+(-\frac{2}{3})^2= \frac{2}{3}+(-\frac{2}{3})^2$

The expression on the Left Hand side is a perfect square.

$(x-\frac{2}{3})^2= \frac{2}{3}+\frac{4}{9}$

$\Rightarrow (x-\frac{2}{3})^2= \frac{10}{9}$

$\Rightarrow (x-\frac{2}{3})=\pm \sqrt{\frac{10}{9}}$

$\Rightarrow (x)=\frac{2}{3} \pm {\frac{\sqrt{10}}{3}$

Splitting the plus or minus sign gives

$x=\frac{2- \sqrt{10}} {3}$

or

$x=\frac{\sqrt{10}+2} {3}$