Solve 1/4x-2x(x+6)=0

$$\frac{ 1 }{ 4 } x-2x(x+6)=0$$

Solve for x

$$x = -\frac{47}{8} = -5\frac{7}{8} = -5.875$$

$$x=0$$

Steps Using Factoring

Multiply -1 and 2 to get -2.

$$\frac{1}{4}x-2x\left(x+6\right)=0$$

Use the distributive property to multiply -2x by x+6.

$$\frac{1}{4}x-2x^{2}-12x=0$$

Combine \frac{1}{4}x and -12x to get -\frac{47}{4}x.

$$-\frac{47}{4}x-2x^{2}=0$$

Factor out x.

$$x\left(-\frac{47}{4}-2x\right)=0$$

To find equation solutions, solve x=0 and -\frac{47}{4}-2x=0.

$$x=0 x=-\frac{47}{8}$$

Steps Using the Quadratic Formula

Multiply -1 and 2 to get -2.

$$\frac{1}{4}x-2x\left(x+6\right)=0$$

Use the distributive property to multiply -2x by x+6.

$$\frac{1}{4}x-2x^{2}-12x=0$$

Combine \frac{1}{4}x and -12x to get -\frac{47}{4}x.

$$-\frac{47}{4}x-2x^{2}=0$$

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.

$$-2x^{2}-\frac{47}{4}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -\frac{47}{4} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{47}{4}\right)±\sqrt{\left(-\frac{47}{4}\right)^{2}}}{2\left(-2\right)}$$

Take the square root of \left(-\frac{47}{4}\right)^{2}.

$$x=\frac{-\left(-\frac{47}{4}\right)±\frac{47}{4}}{2\left(-2\right)}$$

The opposite of -\frac{47}{4} is \frac{47}{4}.

$$x=\frac{\frac{47}{4}±\frac{47}{4}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{\frac{47}{4}±\frac{47}{4}}{-4}$$

Now solve the equation x=\frac{\frac{47}{4}±\frac{47}{4}}{-4} when ± is plus. Add \frac{47}{4} to \frac{47}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{47}{2}}{-4}$$

Divide \frac{47}{2} by -4.

$$x=-\frac{47}{8}$$

Now solve the equation x=\frac{\frac{47}{4}±\frac{47}{4}}{-4} when ± is minus. Subtract \frac{47}{4} from \frac{47}{4} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{0}{-4}$$

Divide 0 by -4.

$$x=0$$

The equation is now solved.

$$x=-\frac{47}{8} x=0$$

Steps for Completing the Square

Multiply -1 and 2 to get -2.

$$\frac{1}{4}x-2x\left(x+6\right)=0$$

Use the distributive property to multiply -2x by x+6.

$$\frac{1}{4}x-2x^{2}-12x=0$$

Combine \frac{1}{4}x and -12x to get -\frac{47}{4}x.

$$-\frac{47}{4}x-2x^{2}=0$$

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.

$$-2x^{2}-\frac{47}{4}x=0$$

Divide both sides by -2.

$$\frac{-2x^{2}-\frac{47}{4}x}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{\frac{47}{4}}{-2}\right)x=\frac{0}{-2}$$

Divide -\frac{47}{4} by -2.

$$x^{2}+\frac{47}{8}x=\frac{0}{-2}$$

Divide 0 by -2.

$$x^{2}+\frac{47}{8}x=0$$

Divide \frac{47}{8}, the coefficient of the x term, by 2 to get \frac{47}{16}. Then add the square of \frac{47}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{47}{8}x+\left(\frac{47}{16}\right)^{2}=\left(\frac{47}{16}\right)^{2}$$

Square \frac{47}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{47}{8}x+\frac{2209}{256}=\frac{2209}{256}$$

Factor x^{2}+\frac{47}{8}x+\frac{2209}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{47}{16}\right)^{2}=\frac{2209}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{47}{16}\right)^{2}}=\sqrt{\frac{2209}{256}}$$

Simplify.

$$x+\frac{47}{16}=\frac{47}{16} x+\frac{47}{16}=-\frac{47}{16}$$

Subtract \frac{47}{16} from both sides of the equation.

$$x=0 x=-\frac{47}{8}$$