Solve 128x^2+384x=124

$$128 x ^ { 2 } + 384 x = 124$$

Solve for x

$$x=\frac{\sqrt{206}}{8}-\frac{3}{2}\approx 0.294087512$$

$$x=-\frac{\sqrt{206}}{8}-\frac{3}{2}\approx -3.294087512$$

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.

$$128x^{2}+384x=124$$

Subtract 124 from both sides of the equation.

$$128x^{2}+384x-124=124-124$$

Subtracting 124 from itself leaves 0.

$$128x^{2}+384x-124=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 128 for a, 384 for b, and -124 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-384±\sqrt{384^{2}-4\times 128\left(-124\right)}}{2\times 128}$$

Square 384.

$$x=\frac{-384±\sqrt{147456-4\times 128\left(-124\right)}}{2\times 128}$$

Multiply -4 times 128.

$$x=\frac{-384±\sqrt{147456-512\left(-124\right)}}{2\times 128}$$

Multiply -512 times -124.

$$x=\frac{-384±\sqrt{147456+63488}}{2\times 128}$$

Add 147456 to 63488.

$$x=\frac{-384±\sqrt{210944}}{2\times 128}$$

Take the square root of 210944.

$$x=\frac{-384±32\sqrt{206}}{2\times 128}$$

Multiply 2 times 128.

$$x=\frac{-384±32\sqrt{206}}{256}$$

Now solve the equation x=\frac{-384±32\sqrt{206}}{256} when ± is plus. Add -384 to 32\sqrt{206}.

$$x=\frac{32\sqrt{206}-384}{256}$$

Divide -384+32\sqrt{206} by 256.

$$x=\frac{\sqrt{206}}{8}-\frac{3}{2}$$

Now solve the equation x=\frac{-384±32\sqrt{206}}{256} when ± is minus. Subtract 32\sqrt{206} from -384.

$$x=\frac{-32\sqrt{206}-384}{256}$$

Divide -384-32\sqrt{206} by 256.

$$x=-\frac{\sqrt{206}}{8}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{206}}{8}-\frac{3}{2} x=-\frac{\sqrt{206}}{8}-\frac{3}{2}$$

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.

$$128x^{2}+384x=124$$

Divide both sides by 128.

$$\frac{128x^{2}+384x}{128}=\frac{124}{128}$$

Dividing by 128 undoes the multiplication by 128.

$$x^{2}+\frac{384}{128}x=\frac{124}{128}$$

Divide 384 by 128.

$$x^{2}+3x=\frac{124}{128}$$

Reduce the fraction \frac{124}{128} to lowest terms by extracting and canceling out 4.

$$x^{2}+3x=\frac{31}{32}$$

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.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{31}{32}+\left(\frac{3}{2}\right)^{2}$$

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

$$x^{2}+3x+\frac{9}{4}=\frac{31}{32}+\frac{9}{4}$$

Add \frac{31}{32} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{103}{32}$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{103}{32}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{103}{32}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{206}}{8} x+\frac{3}{2}=-\frac{\sqrt{206}}{8}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{206}}{8}-\frac{3}{2} x=-\frac{\sqrt{206}}{8}-\frac{3}{2}$$