Solve 9.13=81+x^2-9x

$$9.13 = 81+ { x }^{ 2 } -9x$$

Solve for x (complex solution)

$$x=\frac{i\sqrt{5162}}{10}+4.5\approx 4.5+7.184705979i$$

$$x=-\frac{i\sqrt{5162}}{10}+4.5\approx 4.5-7.184705979i$$

Steps Using the Quadratic Formula

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

$$81+x^{2}-9x=9.13$$

Subtract 9.13 from both sides.

$$81+x^{2}-9x-9.13=0$$

Subtract 9.13 from 81 to get 71.87.

$$71.87+x^{2}-9x=0$$

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.

$$x^{2}-9x+71.87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -9 for b, and 71.87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 71.87}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 71.87}}{2}$$

Multiply -4 times 71.87.

$$x=\frac{-\left(-9\right)±\sqrt{81-287.48}}{2}$$

Add 81 to -287.48.

$$x=\frac{-\left(-9\right)±\sqrt{-206.48}}{2}$$

Take the square root of -206.48.

$$x=\frac{-\left(-9\right)±\frac{\sqrt{5162}i}{5}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\frac{\sqrt{5162}i}{5}}{2}$$

Now solve the equation x=\frac{9±\frac{\sqrt{5162}i}{5}}{2} when ± is plus. Add 9 to \frac{i\sqrt{5162}}{5}.

$$x=\frac{\frac{\sqrt{5162}i}{5}+9}{2}$$

Divide 9+\frac{i\sqrt{5162}}{5} by 2.

$$x=\frac{\sqrt{5162}i}{10}+\frac{9}{2}$$

Now solve the equation x=\frac{9±\frac{\sqrt{5162}i}{5}}{2} when ± is minus. Subtract \frac{i\sqrt{5162}}{5} from 9.

$$x=\frac{-\frac{\sqrt{5162}i}{5}+9}{2}$$

Divide 9-\frac{i\sqrt{5162}}{5} by 2.

$$x=-\frac{\sqrt{5162}i}{10}+\frac{9}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{5162}i}{10}+\frac{9}{2} x=-\frac{\sqrt{5162}i}{10}+\frac{9}{2}$$

Steps for Completing the Square

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

$$81+x^{2}-9x=9.13$$

Subtract 81 from both sides.

$$x^{2}-9x=9.13-81$$

Subtract 81 from 9.13 to get -71.87.

$$x^{2}-9x=-71.87$$

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.

$$x^{2}-9x=-\frac{7187}{100}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-\frac{7187}{100}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=-\frac{7187}{100}+\frac{81}{4}$$

Add -\frac{7187}{100} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-9x+\frac{81}{4}=-\frac{2581}{50}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=-\frac{2581}{50}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{-\frac{2581}{50}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{5162}i}{10} x-\frac{9}{2}=-\frac{\sqrt{5162}i}{10}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{5162}i}{10}+\frac{9}{2} x=-\frac{\sqrt{5162}i}{10}+\frac{9}{2}$$