Solve 192=16x-1/4x^2

$$192=16x- \frac{ 1 }{ 4 } { x }^{ 2 }$$

Solve for x

$$x=16$$

$$x=48$$

Steps Using the Quadratic Formula

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

$$16x-\frac{1}{4}x^{2}=192$$

Subtract 192 from both sides.

$$16x-\frac{1}{4}x^{2}-192=0$$

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.

$$-\frac{1}{4}x^{2}+16x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{4} for a, 16 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-\frac{1}{4}\right)\left(-192\right)}}{2\left(-\frac{1}{4}\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-\frac{1}{4}\right)\left(-192\right)}}{2\left(-\frac{1}{4}\right)}$$

Multiply -4 times -\frac{1}{4}.

$$x=\frac{-16±\sqrt{256-192}}{2\left(-\frac{1}{4}\right)}$$

Add 256 to -192.

$$x=\frac{-16±\sqrt{64}}{2\left(-\frac{1}{4}\right)}$$

Take the square root of 64.

$$x=\frac{-16±8}{2\left(-\frac{1}{4}\right)}$$

Multiply 2 times -\frac{1}{4}.

$$x=\frac{-16±8}{-\frac{1}{2}}$$

Now solve the equation x=\frac{-16±8}{-\frac{1}{2}} when ± is plus. Add -16 to 8.

$$x=-\frac{8}{-\frac{1}{2}}$$

Divide -8 by -\frac{1}{2} by multiplying -8 by the reciprocal of -\frac{1}{2}.

$$x=16$$

Now solve the equation x=\frac{-16±8}{-\frac{1}{2}} when ± is minus. Subtract 8 from -16.

$$x=-\frac{24}{-\frac{1}{2}}$$

Divide -24 by -\frac{1}{2} by multiplying -24 by the reciprocal of -\frac{1}{2}.

$$x=48$$

The equation is now solved.

$$x=16 x=48$$

Steps for Completing the Square

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

$$16x-\frac{1}{4}x^{2}=192$$

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.

$$-\frac{1}{4}x^{2}+16x=192$$

Multiply both sides by -4.

$$\frac{-\frac{1}{4}x^{2}+16x}{-\frac{1}{4}}=\frac{192}{-\frac{1}{4}}$$

Dividing by -\frac{1}{4} undoes the multiplication by -\frac{1}{4}.

$$x^{2}+\frac{16}{-\frac{1}{4}}x=\frac{192}{-\frac{1}{4}}$$

Divide 16 by -\frac{1}{4} by multiplying 16 by the reciprocal of -\frac{1}{4}.

$$x^{2}-64x=\frac{192}{-\frac{1}{4}}$$

Divide 192 by -\frac{1}{4} by multiplying 192 by the reciprocal of -\frac{1}{4}.

$$x^{2}-64x=-768$$

Divide -64, the coefficient of the x term, by 2 to get -32. Then add the square of -32 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-64x+\left(-32\right)^{2}=-768+\left(-32\right)^{2}$$

Square -32.

$$x^{2}-64x+1024=-768+1024$$

Add -768 to 1024.

$$x^{2}-64x+1024=256$$

Factor x^{2}-64x+1024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-32\right)^{2}=256$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-32\right)^{2}}=\sqrt{256}$$

Simplify.

$$x-32=16 x-32=-16$$

Add 32 to both sides of the equation.

$$x=48 x=16$$