Solve 432=192x-16x^2

$$432=192x-16 { x }^{ 2 }$$

Solve for x

$$x=3$$

$$x=9$$

Steps Using the Quadratic Formula

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

$$192x-16x^{2}=432$$

Subtract 432 from both sides.

$$192x-16x^{2}-432=0$$

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.

$$-16x^{2}+192x-432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 192 for b, and -432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-192±\sqrt{192^{2}-4\left(-16\right)\left(-432\right)}}{2\left(-16\right)}$$

Square 192.

$$x=\frac{-192±\sqrt{36864-4\left(-16\right)\left(-432\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-192±\sqrt{36864+64\left(-432\right)}}{2\left(-16\right)}$$

Multiply 64 times -432.

$$x=\frac{-192±\sqrt{36864-27648}}{2\left(-16\right)}$$

Add 36864 to -27648.

$$x=\frac{-192±\sqrt{9216}}{2\left(-16\right)}$$

Take the square root of 9216.

$$x=\frac{-192±96}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-192±96}{-32}$$

Now solve the equation x=\frac{-192±96}{-32} when ± is plus. Add -192 to 96.

$$x=-\frac{96}{-32}$$

Divide -96 by -32.

$$x=3$$

Now solve the equation x=\frac{-192±96}{-32} when ± is minus. Subtract 96 from -192.

$$x=-\frac{288}{-32}$$

Divide -288 by -32.

$$x=9$$

The equation is now solved.

$$x=3 x=9$$

Steps for Completing the Square

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

$$192x-16x^{2}=432$$

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.

$$-16x^{2}+192x=432$$

Divide both sides by -16.

$$\frac{-16x^{2}+192x}{-16}=\frac{432}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{192}{-16}x=\frac{432}{-16}$$

Divide 192 by -16.

$$x^{2}-12x=\frac{432}{-16}$$

Divide 432 by -16.

$$x^{2}-12x=-27$$

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.

$$x^{2}-12x+\left(-6\right)^{2}=-27+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-27+36$$

Add -27 to 36.

$$x^{2}-12x+36=9$$

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}.

$$\left(x-6\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-6=3 x-6=-3$$

Add 6 to both sides of the equation.

$$x=9 x=3$$