Solve 1/4x^2+96x-996=0

$$\frac { 1 } { 4 } x ^ { 2 } + 96 x - 996 = 0$$

Solve for x

$$x=4\sqrt{2553}-192\approx 10.108881547$$

$$x=-4\sqrt{2553}-192\approx -394.108881547$$

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.

$$\frac{1}{4}x^{2}+96x-996=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, 96 for b, and -996 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-96±\sqrt{96^{2}-4\times \frac{1}{4}\left(-996\right)}}{2\times \frac{1}{4}}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\times \frac{1}{4}\left(-996\right)}}{2\times \frac{1}{4}}$$

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

$$x=\frac{-96±\sqrt{9216-\left(-996\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -996.

$$x=\frac{-96±\sqrt{9216+996}}{2\times \frac{1}{4}}$$

Add 9216 to 996.

$$x=\frac{-96±\sqrt{10212}}{2\times \frac{1}{4}}$$

Take the square root of 10212.

$$x=\frac{-96±2\sqrt{2553}}{2\times \frac{1}{4}}$$

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

$$x=\frac{-96±2\sqrt{2553}}{\frac{1}{2}}$$

Now solve the equation x=\frac{-96±2\sqrt{2553}}{\frac{1}{2}} when ± is plus. Add -96 to 2\sqrt{2553}.

$$x=\frac{2\sqrt{2553}-96}{\frac{1}{2}}$$

Divide -96+2\sqrt{2553} by \frac{1}{2} by multiplying -96+2\sqrt{2553} by the reciprocal of \frac{1}{2}.

$$x=4\sqrt{2553}-192$$

Now solve the equation x=\frac{-96±2\sqrt{2553}}{\frac{1}{2}} when ± is minus. Subtract 2\sqrt{2553} from -96.

$$x=\frac{-2\sqrt{2553}-96}{\frac{1}{2}}$$

Divide -96-2\sqrt{2553} by \frac{1}{2} by multiplying -96-2\sqrt{2553} by the reciprocal of \frac{1}{2}.

$$x=-4\sqrt{2553}-192$$

The equation is now solved.

$$x=4\sqrt{2553}-192 x=-4\sqrt{2553}-192$$

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.

$$\frac{1}{4}x^{2}+96x-996=0$$

Add 996 to both sides of the equation.

$$\frac{1}{4}x^{2}+96x-996-\left(-996\right)=-\left(-996\right)$$

Subtracting -996 from itself leaves 0.

$$\frac{1}{4}x^{2}+96x=-\left(-996\right)$$

Subtract -996 from 0.

$$\frac{1}{4}x^{2}+96x=996$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}x^{2}+96x}{\frac{1}{4}}=\frac{996}{\frac{1}{4}}$$

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

$$x^{2}+\frac{96}{\frac{1}{4}}x=\frac{996}{\frac{1}{4}}$$

Divide 96 by \frac{1}{4} by multiplying 96 by the reciprocal of \frac{1}{4}.

$$x^{2}+384x=\frac{996}{\frac{1}{4}}$$

Divide 996 by \frac{1}{4} by multiplying 996 by the reciprocal of \frac{1}{4}.

$$x^{2}+384x=3984$$

Divide 384, the coefficient of the x term, by 2 to get 192. Then add the square of 192 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+384x+192^{2}=3984+192^{2}$$

Square 192.

$$x^{2}+384x+36864=3984+36864$$

Add 3984 to 36864.

$$x^{2}+384x+36864=40848$$

Factor x^{2}+384x+36864. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+192\right)^{2}=40848$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+192\right)^{2}}=\sqrt{40848}$$

Simplify.

$$x+192=4\sqrt{2553} x+192=-4\sqrt{2553}$$

Subtract 192 from both sides of the equation.

$$x=4\sqrt{2553}-192 x=-4\sqrt{2553}-192$$