Solve 2x^2+5=3x+2

$$2 x ^ { 2 } + 5 = 3 x + 2$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{15}i}{4}\approx 0.75+0.968245837i$$

$$x=\frac{-\sqrt{15}i+3}{4}\approx 0.75-0.968245837i$$

Steps Using the Quadratic Formula

Subtract 3x from both sides.

$$2x^{2}+5-3x=2$$

Subtract 2 from both sides.

$$2x^{2}+5-3x-2=0$$

Subtract 2 from 5 to get 3.

$$2x^{2}+3-3x=0$$

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.

$$2x^{2}-3x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\times 3}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\times 3}}{2\times 2}$$

Multiply -8 times 3.

$$x=\frac{-\left(-3\right)±\sqrt{9-24}}{2\times 2}$$

Add 9 to -24.

$$x=\frac{-\left(-3\right)±\sqrt{-15}}{2\times 2}$$

Take the square root of -15.

$$x=\frac{-\left(-3\right)±\sqrt{15}i}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{15}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±\sqrt{15}i}{4}$$

Now solve the equation x=\frac{3±\sqrt{15}i}{4} when ± is plus. Add 3 to i\sqrt{15}.

$$x=\frac{3+\sqrt{15}i}{4}$$

Now solve the equation x=\frac{3±\sqrt{15}i}{4} when ± is minus. Subtract i\sqrt{15} from 3.

$$x=\frac{-\sqrt{15}i+3}{4}$$

The equation is now solved.

$$x=\frac{3+\sqrt{15}i}{4} x=\frac{-\sqrt{15}i+3}{4}$$

Steps for Completing the Square

Subtract 3x from both sides.

$$2x^{2}+5-3x=2$$

Subtract 5 from both sides.

$$2x^{2}-3x=2-5$$

Subtract 5 from 2 to get -3.

$$2x^{2}-3x=-3$$

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{3}{2}+\frac{9}{16}$$

Add -\frac{3}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{15}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=-\frac{15}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{\sqrt{15}i}{4} x-\frac{3}{4}=-\frac{\sqrt{15}i}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=\frac{3+\sqrt{15}i}{4} x=\frac{-\sqrt{15}i+3}{4}$$