Solve 0=.48x^2+384x-12.68

$$0 = .48 x ^ { 2 } + 384 x - 12.68$$

Solve for x

$$x=\frac{\sqrt{5760951}}{6}-400\approx 0.03301947$$

$$x=-\frac{\sqrt{5760951}}{6}-400\approx -800.03301947$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$0.48x^{2}+384x-12.68=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.48 for a, 384 for b, and -12.68 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-384±\sqrt{384^{2}-4\times 0.48\left(-12.68\right)}}{2\times 0.48}$$

Square 384.

$$x=\frac{-384±\sqrt{147456-4\times 0.48\left(-12.68\right)}}{2\times 0.48}$$

Multiply -4 times 0.48.

$$x=\frac{-384±\sqrt{147456-1.92\left(-12.68\right)}}{2\times 0.48}$$

Multiply -1.92 times -12.68 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-384±\sqrt{147456+24.3456}}{2\times 0.48}$$

Add 147456 to 24.3456.

$$x=\frac{-384±\sqrt{147480.3456}}{2\times 0.48}$$

Take the square root of 147480.3456.

$$x=\frac{-384±\frac{4\sqrt{5760951}}{25}}{2\times 0.48}$$

Multiply 2 times 0.48.

$$x=\frac{-384±\frac{4\sqrt{5760951}}{25}}{0.96}$$

Now solve the equation x=\frac{-384±\frac{4\sqrt{5760951}}{25}}{0.96} when ± is plus. Add -384 to \frac{4\sqrt{5760951}}{25}.

$$x=\frac{\frac{4\sqrt{5760951}}{25}-384}{0.96}$$

Divide -384+\frac{4\sqrt{5760951}}{25} by 0.96 by multiplying -384+\frac{4\sqrt{5760951}}{25} by the reciprocal of 0.96.

$$x=\frac{\sqrt{5760951}}{6}-400$$

Now solve the equation x=\frac{-384±\frac{4\sqrt{5760951}}{25}}{0.96} when ± is minus. Subtract \frac{4\sqrt{5760951}}{25} from -384.

$$x=\frac{-\frac{4\sqrt{5760951}}{25}-384}{0.96}$$

Divide -384-\frac{4\sqrt{5760951}}{25} by 0.96 by multiplying -384-\frac{4\sqrt{5760951}}{25} by the reciprocal of 0.96.

$$x=-\frac{\sqrt{5760951}}{6}-400$$

The equation is now solved.

$$x=\frac{\sqrt{5760951}}{6}-400 x=-\frac{\sqrt{5760951}}{6}-400$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$0.48x^{2}+384x-12.68=0$$

Add 12.68 to both sides. Anything plus zero gives itself.

$$0.48x^{2}+384x=12.68$$

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.

$$0.48x^{2}+384x=\frac{317}{25}$$

Divide both sides of the equation by 0.48, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.48x^{2}+384x}{0.48}=\frac{\frac{317}{25}}{0.48}$$

Dividing by 0.48 undoes the multiplication by 0.48.

$$x^{2}+\frac{384}{0.48}x=\frac{\frac{317}{25}}{0.48}$$

Divide 384 by 0.48 by multiplying 384 by the reciprocal of 0.48.

$$x^{2}+800x=\frac{\frac{317}{25}}{0.48}$$

Divide \frac{317}{25} by 0.48 by multiplying \frac{317}{25} by the reciprocal of 0.48.

$$x^{2}+800x=\frac{317}{12}$$

Divide 800, the coefficient of the x term, by 2 to get 400. Then add the square of 400 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+800x+400^{2}=\frac{317}{12}+400^{2}$$

Square 400.

$$x^{2}+800x+160000=\frac{317}{12}+160000$$

Add \frac{317}{12} to 160000.

$$x^{2}+800x+160000=\frac{1920317}{12}$$

Factor x^{2}+800x+160000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+400\right)^{2}=\frac{1920317}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+400\right)^{2}}=\sqrt{\frac{1920317}{12}}$$

Simplify.

$$x+400=\frac{\sqrt{5760951}}{6} x+400=-\frac{\sqrt{5760951}}{6}$$

Subtract 400 from both sides of the equation.

$$x=\frac{\sqrt{5760951}}{6}-400 x=-\frac{\sqrt{5760951}}{6}-400$$