Solve 72x-6x^2=216

$$72 x - 6 x ^ { 2 } = 216$$

Solve for x

$$x=6$$

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.

$$-6x^{2}+72x=216$$

Subtract 216 from both sides of the equation.

$$-6x^{2}+72x-216=216-216$$

Subtracting 216 from itself leaves 0.

$$-6x^{2}+72x-216=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 72 for b, and -216 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-6\right)\left(-216\right)}}{2\left(-6\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-6\right)\left(-216\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-72±\sqrt{5184+24\left(-216\right)}}{2\left(-6\right)}$$

Multiply 24 times -216.

$$x=\frac{-72±\sqrt{5184-5184}}{2\left(-6\right)}$$

Add 5184 to -5184.

$$x=\frac{-72±\sqrt{0}}{2\left(-6\right)}$$

Take the square root of 0.

$$x=-\frac{72}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=-\frac{72}{-12}$$

Divide -72 by -12.

$$x=6$$

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.

$$-6x^{2}+72x=216$$

Divide both sides by -6.

$$\frac{-6x^{2}+72x}{-6}=\frac{216}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{72}{-6}x=\frac{216}{-6}$$

Divide 72 by -6.

$$x^{2}-12x=\frac{216}{-6}$$

Divide 216 by -6.

$$x^{2}-12x=-36$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-36+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-36+36$$

Add -36 to 36.

$$x^{2}-12x+36=0$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-6=0 x-6=0$$

Add 6 to both sides of the equation.

$$x=6 x=6$$

The equation is now solved. Solutions are the same.

$$x=6$$