Solve 256=96x-16x^2

$$256=96x-16 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=3+\sqrt{7}i\approx 3+2.645751311i$$

$$x=-\sqrt{7}i+3\approx 3-2.645751311i$$

Steps Using the Quadratic Formula

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

$$96x-16x^{2}=256$$

Subtract 256 from both sides.

$$96x-16x^{2}-256=0$$

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.

$$-16x^{2}+96x-256=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 96 for b, and -256 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-96±\sqrt{96^{2}-4\left(-16\right)\left(-256\right)}}{2\left(-16\right)}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\left(-16\right)\left(-256\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-96±\sqrt{9216+64\left(-256\right)}}{2\left(-16\right)}$$

Multiply 64 times -256.

$$x=\frac{-96±\sqrt{9216-16384}}{2\left(-16\right)}$$

Add 9216 to -16384.

$$x=\frac{-96±\sqrt{-7168}}{2\left(-16\right)}$$

Take the square root of -7168.

$$x=\frac{-96±32\sqrt{7}i}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-96±32\sqrt{7}i}{-32}$$

Now solve the equation x=\frac{-96±32\sqrt{7}i}{-32} when ± is plus. Add -96 to 32i\sqrt{7}.

$$x=\frac{-96+32\sqrt{7}i}{-32}$$

Divide -96+32i\sqrt{7} by -32.

$$x=-\sqrt{7}i+3$$

Now solve the equation x=\frac{-96±32\sqrt{7}i}{-32} when ± is minus. Subtract 32i\sqrt{7} from -96.

$$x=\frac{-32\sqrt{7}i-96}{-32}$$

Divide -96-32i\sqrt{7} by -32.

$$x=3+\sqrt{7}i$$

The equation is now solved.

$$x=-\sqrt{7}i+3 x=3+\sqrt{7}i$$

Steps for Completing the Square

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

$$96x-16x^{2}=256$$

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.

$$-16x^{2}+96x=256$$

Divide both sides by -16.

$$\frac{-16x^{2}+96x}{-16}=\frac{256}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{96}{-16}x=\frac{256}{-16}$$

Divide 96 by -16.

$$x^{2}-6x=\frac{256}{-16}$$

Divide 256 by -16.

$$x^{2}-6x=-16$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-16+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-16+9$$

Add -16 to 9.

$$x^{2}-6x+9=-7$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=-7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-7}$$

Simplify.

$$x-3=\sqrt{7}i x-3=-\sqrt{7}i$$

Add 3 to both sides of the equation.

$$x=3+\sqrt{7}i x=-\sqrt{7}i+3$$