Solve 1x^2=-10x-8

$$1 x ^ { 2 } = - 10 x - 8$$

Solve for x (complex solution)

$$x=\sqrt{17}-5\approx -0.876894374$$

$$x=-\left(\sqrt{17}+5\right)\approx -9.123105626$$

Steps Using the Quadratic Formula

Add 10x to both sides.

$$1x^{2}+10x=-8$$

Add 8 to both sides.

$$1x^{2}+10x+8=0$$

Reorder the terms.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-10±\sqrt{100-32}}{2}$$

Add 100 to -32.

$$x=\frac{-10±\sqrt{68}}{2}$$

Take the square root of 68.

$$x=\frac{-10±2\sqrt{17}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{2} when ± is plus. Add -10 to 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-10}{2}$$

Divide -10+2\sqrt{17} by 2.

$$x=\sqrt{17}-5$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{2} when ± is minus. Subtract 2\sqrt{17} from -10.

$$x=\frac{-2\sqrt{17}-10}{2}$$

Divide -10-2\sqrt{17} by 2.

$$x=-\sqrt{17}-5$$

The equation is now solved.

$$x=\sqrt{17}-5 x=-\sqrt{17}-5$$

Steps for Completing the Square

Add 10x to both sides.

$$1x^{2}+10x=-8$$

Reorder the terms.

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

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

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

Square 5.

$$x^{2}+10x+25=-8+25$$

Add -8 to 25.

$$x^{2}+10x+25=17$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{17}$$

Simplify.

$$x+5=\sqrt{17} x+5=-\sqrt{17}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{17}-5 x=-\sqrt{17}-5$$

Solve for x

$$x=\sqrt{17}-5\approx -0.876894374$$

$$x=-\sqrt{17}-5\approx -9.123105626$$

Steps Using the Quadratic Formula

Add 10x to both sides.

$$1x^{2}+10x=-8$$

Add 8 to both sides.

$$1x^{2}+10x+8=0$$

Reorder the terms.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 8}}{2}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 8}}{2}$$

Multiply -4 times 8.

$$x=\frac{-10±\sqrt{100-32}}{2}$$

Add 100 to -32.

$$x=\frac{-10±\sqrt{68}}{2}$$

Take the square root of 68.

$$x=\frac{-10±2\sqrt{17}}{2}$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{2} when ± is plus. Add -10 to 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-10}{2}$$

Divide -10+2\sqrt{17} by 2.

$$x=\sqrt{17}-5$$

Now solve the equation x=\frac{-10±2\sqrt{17}}{2} when ± is minus. Subtract 2\sqrt{17} from -10.

$$x=\frac{-2\sqrt{17}-10}{2}$$

Divide -10-2\sqrt{17} by 2.

$$x=-\sqrt{17}-5$$

The equation is now solved.

$$x=\sqrt{17}-5 x=-\sqrt{17}-5$$

Steps for Completing the Square

Add 10x to both sides.

$$1x^{2}+10x=-8$$

Reorder the terms.

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

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

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

Square 5.

$$x^{2}+10x+25=-8+25$$

Add -8 to 25.

$$x^{2}+10x+25=17$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=17$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{17}$$

Simplify.

$$x+5=\sqrt{17} x+5=-\sqrt{17}$$

Subtract 5 from both sides of the equation.

$$x=\sqrt{17}-5 x=-\sqrt{17}-5$$