Solve 15x^2-20x+30=0

$$15 { x }^{ 2 } -20x+30=0$$

Solve for x (complex solution)

$$x=\frac{2+\sqrt{14}i}{3}\approx 0.666666667+1.247219129i$$

$$x=\frac{-\sqrt{14}i+2}{3}\approx 0.666666667-1.247219129i$$

Steps Using the Quadratic Formula

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.

$$15x^{2}-20x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -20 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 15\times 30}}{2\times 15}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 15\times 30}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-20\right)±\sqrt{400-60\times 30}}{2\times 15}$$

Multiply -60 times 30.

$$x=\frac{-\left(-20\right)±\sqrt{400-1800}}{2\times 15}$$

Add 400 to -1800.

$$x=\frac{-\left(-20\right)±\sqrt{-1400}}{2\times 15}$$

Take the square root of -1400.

$$x=\frac{-\left(-20\right)±10\sqrt{14}i}{2\times 15}$$

The opposite of -20 is 20.

$$x=\frac{20±10\sqrt{14}i}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{20±10\sqrt{14}i}{30}$$

Now solve the equation x=\frac{20±10\sqrt{14}i}{30} when ± is plus. Add 20 to 10i\sqrt{14}.

$$x=\frac{20+10\sqrt{14}i}{30}$$

Divide 20+10i\sqrt{14} by 30.

$$x=\frac{2+\sqrt{14}i}{3}$$

Now solve the equation x=\frac{20±10\sqrt{14}i}{30} when ± is minus. Subtract 10i\sqrt{14} from 20.

$$x=\frac{-10\sqrt{14}i+20}{30}$$

Divide 20-10i\sqrt{14} by 30.

$$x=\frac{-\sqrt{14}i+2}{3}$$

The equation is now solved.

$$x=\frac{2+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i+2}{3}$$

Steps for Completing the Square

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.

$$15x^{2}-20x+30=0$$

Subtract 30 from both sides of the equation.

$$15x^{2}-20x+30-30=-30$$

Subtracting 30 from itself leaves 0.

$$15x^{2}-20x=-30$$

Divide both sides by 15.

$$\frac{15x^{2}-20x}{15}=-\frac{30}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\left(-\frac{20}{15}\right)x=-\frac{30}{15}$$

Reduce the fraction \frac{-20}{15} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{4}{3}x=-\frac{30}{15}$$

Divide -30 by 15.

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

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

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

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

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

Add -2 to \frac{4}{9}.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=-\frac{14}{9}$$

Factor x^{2}-\frac{4}{3}x+\frac{4}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{3}\right)^{2}=-\frac{14}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{-\frac{14}{9}}$$

Simplify.

$$x-\frac{2}{3}=\frac{\sqrt{14}i}{3} x-\frac{2}{3}=-\frac{\sqrt{14}i}{3}$$

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

$$x=\frac{2+\sqrt{14}i}{3} x=\frac{-\sqrt{14}i+2}{3}$$