Solve 15x^2-5+30x=0

$$15 x ^ { 2 } - 5 + 30 x = 0$$

Solve for x

$$x=\frac{2\sqrt{3}}{3}-1\approx 0.154700538$$

$$x=-\frac{2\sqrt{3}}{3}-1\approx -2.154700538$$

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.

$$15x^{2}+30x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 30 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 15\left(-5\right)}}{2\times 15}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 15\left(-5\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-30±\sqrt{900-60\left(-5\right)}}{2\times 15}$$

Multiply -60 times -5.

$$x=\frac{-30±\sqrt{900+300}}{2\times 15}$$

Add 900 to 300.

$$x=\frac{-30±\sqrt{1200}}{2\times 15}$$

Take the square root of 1200.

$$x=\frac{-30±20\sqrt{3}}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-30±20\sqrt{3}}{30}$$

Now solve the equation x=\frac{-30±20\sqrt{3}}{30} when ± is plus. Add -30 to 20\sqrt{3}.

$$x=\frac{20\sqrt{3}-30}{30}$$

Divide -30+20\sqrt{3} by 30.

$$x=\frac{2\sqrt{3}}{3}-1$$

Now solve the equation x=\frac{-30±20\sqrt{3}}{30} when ± is minus. Subtract 20\sqrt{3} from -30.

$$x=\frac{-20\sqrt{3}-30}{30}$$

Divide -30-20\sqrt{3} by 30.

$$x=-\frac{2\sqrt{3}}{3}-1$$

The equation is now solved.

$$x=\frac{2\sqrt{3}}{3}-1 x=-\frac{2\sqrt{3}}{3}-1$$

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.

$$15x^{2}+30x-5=0$$

Add 5 to both sides of the equation.

$$15x^{2}+30x-5-\left(-5\right)=-\left(-5\right)$$

Subtracting -5 from itself leaves 0.

$$15x^{2}+30x=-\left(-5\right)$$

Subtract -5 from 0.

$$15x^{2}+30x=5$$

Divide both sides by 15.

$$\frac{15x^{2}+30x}{15}=\frac{5}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\frac{30}{15}x=\frac{5}{15}$$

Divide 30 by 15.

$$x^{2}+2x=\frac{5}{15}$$

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

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

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.

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

Square 1.

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

Add \frac{1}{3} to 1.

$$x^{2}+2x+1=\frac{4}{3}$$

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}.

$$\left(x+1\right)^{2}=\frac{4}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{4}{3}}$$

Simplify.

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

Subtract 1 from both sides of the equation.

$$x=\frac{2\sqrt{3}}{3}-1 x=-\frac{2\sqrt{3}}{3}-1$$