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

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

Solve for x (complex solution)

$$x=\frac{\sqrt{6}i}{2}+1\approx 1+1.224744871i$$

$$x=-\frac{\sqrt{6}i}{2}+1\approx 1-1.224744871i$$

Steps Using the Quadratic Formula

Subtract 9x from both sides.

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

Combine 5x and -9x to get -4x.

$$2x^{2}-4x=-5$$

Add 5 to both sides.

$$2x^{2}-4x+5=0$$

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

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

Square -4.

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

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\times 5}}{2\times 2}$$

Multiply -8 times 5.

$$x=\frac{-\left(-4\right)±\sqrt{16-40}}{2\times 2}$$

Add 16 to -40.

$$x=\frac{-\left(-4\right)±\sqrt{-24}}{2\times 2}$$

Take the square root of -24.

$$x=\frac{-\left(-4\right)±2\sqrt{6}i}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±2\sqrt{6}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±2\sqrt{6}i}{4}$$

Now solve the equation x=\frac{4±2\sqrt{6}i}{4} when ± is plus. Add 4 to 2i\sqrt{6}.

$$x=\frac{4+2\sqrt{6}i}{4}$$

Divide 4+2i\sqrt{6} by 4.

$$x=\frac{\sqrt{6}i}{2}+1$$

Now solve the equation x=\frac{4±2\sqrt{6}i}{4} when ± is minus. Subtract 2i\sqrt{6} from 4.

$$x=\frac{-2\sqrt{6}i+4}{4}$$

Divide 4-2i\sqrt{6} by 4.

$$x=-\frac{\sqrt{6}i}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{6}i}{2}+1 x=-\frac{\sqrt{6}i}{2}+1$$

Steps for Completing the Square

Subtract 9x from both sides.

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

Combine 5x and -9x to get -4x.

$$2x^{2}-4x=-5$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=-\frac{5}{2}$$

Dividing by 2 undoes the multiplication by 2.

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

Divide -4 by 2.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=-\frac{5}{2}+1$$

Add -\frac{5}{2} to 1.

$$x^{2}-2x+1=-\frac{3}{2}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=-\frac{3}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{3}{2}}$$

Simplify.

$$x-1=\frac{\sqrt{6}i}{2} x-1=-\frac{\sqrt{6}i}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{6}i}{2}+1 x=-\frac{\sqrt{6}i}{2}+1$$