C
10 million points

Solve the following equation $5x^2-6x-2=0$

chegendungu

C
10 million points

$5x^2-6x-2=0$

Add $2$ to both sides,

$5x^2-6x=2$

Divide through by $5$

$x^2-\frac{6}{5}x=\frac{2}{5}$

Complete the square, by adding $\bigg(\frac{3}{5}\bigg)^2$ to both sides,

$x^2-\bigg(\frac{6}{5}\bigg)x+\bigg(\frac{3}{5}\bigg)^2=\bigg(\frac{3}{5}\bigg)^2+\bigg(\frac{2}{5}\bigg)=\frac{9+10}{25}=\frac{19}{25}$

Factorise the left-hand side,

$\bigg(x-\frac{3}{5}\bigg)^2=\frac{19}{25}$

Take the square root of both sides,

$\bigg(x-\frac{3}{5}\bigg)^2=\pm\sqrt{\frac{19}{25}}=\pm\frac{\sqrt{19}}{5}$

And finally, add $\frac{3}{5}$ to both sides,

$x=\frac{3}{5}\pm\frac{\sqrt{19}}{5}=\frac{3\pm\sqrt{19}}{5}$

[Answers to such questions should be left with surds in them, unless a specified degree of accuracy is demanded by the question.]

Notice that the root can be used to find the factors of the original expression.

$5x^2-6x-2=5(x^2-\frac{6}{5}x-2)$

$=5\bigg(x-\frac{3+\sqrt{19}}{5}\bigg)\bigg(x-\frac{3-\sqrt{19}}{5}\bigg)$

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