Respuesta :

kudzordzifrancis
ANSWER

[tex]x=\frac{2-\sqrt{10}} {3}[/tex]

or

[tex]x=\frac{\sqrt{10}+2} {3}[/tex]

We have

[tex]3x^2-4x-2=0[/tex]

Since we cannot factor easily, we complete the square.

Adding 2 to both sides give,

[tex]3x^2-4x=2[/tex]

Dividing through by 3 gives

[tex]x^2-\frac{4}{3}x= \frac{2}{3}[/tex]

Adding [tex](-\frac{2}{3})^2[/tex] to both sides gives

[tex]x^2-\frac{4}{3}x+(-\frac{2}{3})^2= \frac{2}{3}+(-\frac{2}{3})^2[/tex]

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

[tex](x-\frac{2}{3})^2= \frac{2}{3}+\frac{4}{9}[/tex]

[tex]\Rightarrow (x-\frac{2}{3})^2= \frac{10}{9}[/tex]

[tex]\Rightarrow (x-\frac{2}{3})=\pm \sqrt{\frac{10}{9}}[/tex]

[tex]\Rightarrow (x)=\frac{2}{3} \pm {\frac{\sqrt{10}}{3}[/tex]

Splitting the plus or minus sign gives

[tex]x=\frac{2- \sqrt{10}} {3}[/tex]

or

[tex]x=\frac{\sqrt{10}+2} {3}[/tex]
alinakincsem

Answer:

[tex]x = \frac{2}{3} + \frac{\sqrt{10} }{3} , x = \frac{2}{3} - \frac{\sqrt{10} }{3}[/tex]

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:

[tex]x^{2} - \frac{4}{3} x - \frac{2}{3} = 0[/tex]

[tex]x^{2} - \frac{4}{3} x = \frac{2}{3}[/tex]

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

[tex]x^{2} - \frac{4}{3} x + (-\frac{2}{3} )^{2} = \frac{2}{3} +  (-\frac{2}{3} )^{2}[/tex]

[tex](x - \frac{2}{3} )^2 = \frac{2}{3} + \frac{4}{9}[/tex]

[tex](x - \frac{2}{3} )^2 = \frac{4}{9}[/tex]

[tex]\sqrt{(x - \frac{2}{3} )^2} = \sqrt{\frac{4}{9} }[/tex]

[tex]x - \frac{2}{3} = \sqrt{\frac{10}{9} }[/tex] , [tex]x - \frac{2}{3} = -\sqrt{\frac{10}{9} }[/tex]

[tex]x = \frac{2}{3} + \frac{\sqrt{10} }{3} , x = \frac{2}{3} - \frac{\sqrt{10} }{3}[/tex]



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