Solve 9.13=81+x^2-8x

$$9.13 = 81 + x ^ { 2 } - 8 x$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{5587}}{10}+4\approx 4+7.474623736i$$

$$x=-\frac{i\sqrt{5587}}{10}+4\approx 4-7.474623736i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$81+x^{2}-8x=9.13$$

Subtract 9.13 from both sides.

$$81+x^{2}-8x-9.13=0$$

Subtract 9.13 from 81 to get 71.87.

$$71.87+x^{2}-8x=0$$

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.

$$x^{2}-8x+71.87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and 71.87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 71.87}}{2}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 71.87}}{2}$$

Multiply -4 times 71.87.

$$x=\frac{-\left(-8\right)±\sqrt{64-287.48}}{2}$$

Add 64 to -287.48.

$$x=\frac{-\left(-8\right)±\sqrt{-223.48}}{2}$$

Take the square root of -223.48.

$$x=\frac{-\left(-8\right)±\frac{\sqrt{5587}i}{5}}{2}$$

The opposite of -8 is 8.

$$x=\frac{8±\frac{\sqrt{5587}i}{5}}{2}$$

Now solve the equation x=\frac{8±\frac{\sqrt{5587}i}{5}}{2} when ± is plus. Add 8 to \frac{i\sqrt{5587}}{5}.

$$x=\frac{\frac{\sqrt{5587}i}{5}+8}{2}$$

Divide 8+\frac{i\sqrt{5587}}{5} by 2.

$$x=\frac{\sqrt{5587}i}{10}+4$$

Now solve the equation x=\frac{8±\frac{\sqrt{5587}i}{5}}{2} when ± is minus. Subtract \frac{i\sqrt{5587}}{5} from 8.

$$x=\frac{-\frac{\sqrt{5587}i}{5}+8}{2}$$

Divide 8-\frac{i\sqrt{5587}}{5} by 2.

$$x=-\frac{\sqrt{5587}i}{10}+4$$

The equation is now solved.

$$x=\frac{\sqrt{5587}i}{10}+4 x=-\frac{\sqrt{5587}i}{10}+4$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$81+x^{2}-8x=9.13$$

Subtract 81 from both sides.

$$x^{2}-8x=9.13-81$$

Subtract 81 from 9.13 to get -71.87.

$$x^{2}-8x=-71.87$$

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.

$$x^{2}-8x=-\frac{7187}{100}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{7187}{100}+\left(-4\right)^{2}$$

Square -4.

$$x^{2}-8x+16=-\frac{7187}{100}+16$$

Add -\frac{7187}{100} to 16.

$$x^{2}-8x+16=-\frac{5587}{100}$$

Factor x^{2}-8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-4\right)^{2}=-\frac{5587}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-4\right)^{2}}=\sqrt{-\frac{5587}{100}}$$

Simplify.

$$x-4=\frac{\sqrt{5587}i}{10} x-4=-\frac{\sqrt{5587}i}{10}$$

Add 4 to both sides of the equation.

$$x=\frac{\sqrt{5587}i}{10}+4 x=-\frac{\sqrt{5587}i}{10}+4$$