Solve 9^2+2x^2=5x

$$9 ^ { 2 } + 2 x ^ { 2 } = 5 x$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{623}i}{4}\approx 1.25+6.239991987i$$

$$x=\frac{-\sqrt{623}i+5}{4}\approx 1.25-6.239991987i$$

Steps Using the Quadratic Formula

Calculate 9 to the power of 2 and get 81.

$$81+2x^{2}=5x$$

Subtract 5x from both sides.

$$81+2x^{2}-5x=0$$

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.

$$2x^{2}-5x+81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and 81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\times 81}}{2\times 2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\times 81}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\times 81}}{2\times 2}$$

Multiply -8 times 81.

$$x=\frac{-\left(-5\right)±\sqrt{25-648}}{2\times 2}$$

Add 25 to -648.

$$x=\frac{-\left(-5\right)±\sqrt{-623}}{2\times 2}$$

Take the square root of -623.

$$x=\frac{-\left(-5\right)±\sqrt{623}i}{2\times 2}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{623}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5±\sqrt{623}i}{4}$$

Now solve the equation x=\frac{5±\sqrt{623}i}{4} when ± is plus. Add 5 to i\sqrt{623}.

$$x=\frac{5+\sqrt{623}i}{4}$$

Now solve the equation x=\frac{5±\sqrt{623}i}{4} when ± is minus. Subtract i\sqrt{623} from 5.

$$x=\frac{-\sqrt{623}i+5}{4}$$

The equation is now solved.

$$x=\frac{5+\sqrt{623}i}{4} x=\frac{-\sqrt{623}i+5}{4}$$

Steps for Completing the Square

Calculate 9 to the power of 2 and get 81.

$$81+2x^{2}=5x$$

Subtract 5x from both sides.

$$81+2x^{2}-5x=0$$

Subtract 81 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-5x=-81$$

Divide both sides by 2.

$$\frac{2x^{2}-5x}{2}=-\frac{81}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{5}{2}x=-\frac{81}{2}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=-\frac{81}{2}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{81}{2}+\frac{25}{16}$$

Add -\frac{81}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{623}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{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}.

$$\left(x-\frac{5}{4}\right)^{2}=-\frac{623}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{623}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{\sqrt{623}i}{4} x-\frac{5}{4}=-\frac{\sqrt{623}i}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$x=\frac{5+\sqrt{623}i}{4} x=\frac{-\sqrt{623}i+5}{4}$$