Solve 2x(x-3)=-15

$$2 x ( x - 3 ) = - 15$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{21}i}{2}\approx 1.5+2.291287847i$$

$$x=\frac{-\sqrt{21}i+3}{2}\approx 1.5-2.291287847i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by x-3.

$$2x^{2}-6x=-15$$

Add 15 to both sides.

$$2x^{2}-6x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -6 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\times 15}}{2\times 2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\times 15}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\times 15}}{2\times 2}$$

Multiply -8 times 15.

$$x=\frac{-\left(-6\right)±\sqrt{36-120}}{2\times 2}$$

Add 36 to -120.

$$x=\frac{-\left(-6\right)±\sqrt{-84}}{2\times 2}$$

Take the square root of -84.

$$x=\frac{-\left(-6\right)±2\sqrt{21}i}{2\times 2}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{21}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{6±2\sqrt{21}i}{4}$$

Now solve the equation x=\frac{6±2\sqrt{21}i}{4} when ± is plus. Add 6 to 2i\sqrt{21}.

$$x=\frac{6+2\sqrt{21}i}{4}$$

Divide 6+2i\sqrt{21} by 4.

$$x=\frac{3+\sqrt{21}i}{2}$$

Now solve the equation x=\frac{6±2\sqrt{21}i}{4} when ± is minus. Subtract 2i\sqrt{21} from 6.

$$x=\frac{-2\sqrt{21}i+6}{4}$$

Divide 6-2i\sqrt{21} by 4.

$$x=\frac{-\sqrt{21}i+3}{2}$$

The equation is now solved.

$$x=\frac{3+\sqrt{21}i}{2} x=\frac{-\sqrt{21}i+3}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 2x by x-3.

$$2x^{2}-6x=-15$$

Divide both sides by 2.

$$\frac{2x^{2}-6x}{2}=-\frac{15}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{6}{2}\right)x=-\frac{15}{2}$$

Divide -6 by 2.

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

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

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-\frac{15}{2}+\frac{9}{4}$$

Add -\frac{15}{2} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3x+\frac{9}{4}=-\frac{21}{4}$$

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

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{21}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{21}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{21}i}{2} x-\frac{3}{2}=-\frac{\sqrt{21}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{3+\sqrt{21}i}{2} x=\frac{-\sqrt{21}i+3}{2}$$