Solve 108x-4x^2=0

$$108 x - 4 x ^ { 2 } = 0$$

Solve for x

$$x=27$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(108-4x\right)=0$$

To find equation solutions, solve x=0 and 108-4x=0.

$$x=0 x=27$$

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.

$$-4x^{2}+108x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 108 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}}}{2\left(-4\right)}$$

Take the square root of 108^{2}.

$$x=\frac{-108±108}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-108±108}{-8}$$

Now solve the equation x=\frac{-108±108}{-8} when ± is plus. Add -108 to 108.

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

Divide 0 by -8.

$$x=0$$

Now solve the equation x=\frac{-108±108}{-8} when ± is minus. Subtract 108 from -108.

$$x=-\frac{216}{-8}$$

Divide -216 by -8.

$$x=27$$

The equation is now solved.

$$x=0 x=27$$

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.

$$-4x^{2}+108x=0$$

Divide both sides by -4.

$$\frac{-4x^{2}+108x}{-4}=\frac{0}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{108}{-4}x=\frac{0}{-4}$$

Divide 108 by -4.

$$x^{2}-27x=\frac{0}{-4}$$

Divide 0 by -4.

$$x^{2}-27x=0$$

Divide -27, the coefficient of the x term, by 2 to get -\frac{27}{2}. Then add the square of -\frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-27x+\left(-\frac{27}{2}\right)^{2}=\left(-\frac{27}{2}\right)^{2}$$

Square -\frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-27x+\frac{729}{4}=\frac{729}{4}$$

Factor x^{2}-27x+\frac{729}{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}.

$$\left(x-\frac{27}{2}\right)^{2}=\frac{729}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Simplify.

$$x-\frac{27}{2}=\frac{27}{2} x-\frac{27}{2}=-\frac{27}{2}$$

Add \frac{27}{2} to both sides of the equation.

$$x=27 x=0$$