Solve x-5x^2-8x+10=0

$$x - 5 x ^ { 2 } - 8 x + 10 = 0$$

Solve for x

$$x=\frac{\sqrt{249}-7}{10}\approx 0.877973384$$

$$x=\frac{-\sqrt{249}-7}{10}\approx -2.277973384$$

Steps Using the Quadratic Formula

Combine x and -8x to get -7x.

$$-7x-5x^{2}+10=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-5x^{2}-7x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -7 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-5\right)\times 10}}{2\left(-5\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-5\right)\times 10}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-\left(-7\right)±\sqrt{49+20\times 10}}{2\left(-5\right)}$$

Multiply 20 times 10.

$$x=\frac{-\left(-7\right)±\sqrt{49+200}}{2\left(-5\right)}$$

Add 49 to 200.

$$x=\frac{-\left(-7\right)±\sqrt{249}}{2\left(-5\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{249}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{7±\sqrt{249}}{-10}$$

Now solve the equation x=\frac{7±\sqrt{249}}{-10} when ± is plus. Add 7 to \sqrt{249}.

$$x=\frac{\sqrt{249}+7}{-10}$$

Divide 7+\sqrt{249} by -10.

$$x=\frac{-\sqrt{249}-7}{10}$$

Now solve the equation x=\frac{7±\sqrt{249}}{-10} when ± is minus. Subtract \sqrt{249} from 7.

$$x=\frac{7-\sqrt{249}}{-10}$$

Divide 7-\sqrt{249} by -10.

$$x=\frac{\sqrt{249}-7}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{249}-7}{10} x=\frac{\sqrt{249}-7}{10}$$

Steps for Completing the Square

Combine x and -8x to get -7x.

$$-7x-5x^{2}+10=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$-7x-5x^{2}=-10$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-5x^{2}-7x=-10$$

Divide both sides by -5.

$$\frac{-5x^{2}-7x}{-5}=-\frac{10}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\left(-\frac{7}{-5}\right)x=-\frac{10}{-5}$$

Divide -7 by -5.

$$x^{2}+\frac{7}{5}x=-\frac{10}{-5}$$

Divide -10 by -5.

$$x^{2}+\frac{7}{5}x=2$$

Divide \frac{7}{5}, the coefficient of the x term, by 2 to get \frac{7}{10}. Then add the square of \frac{7}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{5}x+\left(\frac{7}{10}\right)^{2}=2+\left(\frac{7}{10}\right)^{2}$$

Square \frac{7}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=2+\frac{49}{100}$$

Add 2 to \frac{49}{100}.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=\frac{249}{100}$$

Factor x^{2}+\frac{7}{5}x+\frac{49}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{10}\right)^{2}=\frac{249}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{10}\right)^{2}}=\sqrt{\frac{249}{100}}$$

Simplify.

$$x+\frac{7}{10}=\frac{\sqrt{249}}{10} x+\frac{7}{10}=-\frac{\sqrt{249}}{10}$$

Subtract \frac{7}{10} from both sides of the equation.

$$x=\frac{\sqrt{249}-7}{10} x=\frac{-\sqrt{249}-7}{10}$$