Solve -64c^2-2304c+36864=0

$$-64 { c }^{ 2 } -2304c+36864=0$$

Solve for c

$$c=-48$$

$$c=12$$

Steps Using Factoring By Grouping

Divide both sides by 64.

$$-c^{2}-36c+576=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -c^{2}+ac+bc+576. To find a and b, set up a system to be solved.

$$a+b=-36 ab=-576=-576$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -576.

$$1,-576 2,-288 3,-192 4,-144 6,-96 8,-72 9,-64 12,-48 16,-36 18,-32 24,-24$$

Calculate the sum for each pair.

$$1-576=-575 2-288=-286 3-192=-189 4-144=-140 6-96=-90 8-72=-64 9-64=-55 12-48=-36 16-36=-20 18-32=-14 24-24=0$$

The solution is the pair that gives sum -36.

$$a=12 b=-48$$

Rewrite -c^{2}-36c+576 as \left(-c^{2}+12c\right)+\left(-48c+576\right).

$$\left(-c^{2}+12c\right)+\left(-48c+576\right)$$

Factor out c in the first and 48 in the second group.

$$c\left(-c+12\right)+48\left(-c+12\right)$$

Factor out common term -c+12 by using distributive property.

$$\left(-c+12\right)\left(c+48\right)$$

To find equation solutions, solve -c+12=0 and c+48=0.

$$c=12 c=-48$$

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.

$$-64c^{2}-2304c+36864=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -64 for a, -2304 for b, and 36864 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-2304\right)±\sqrt{\left(-2304\right)^{2}-4\left(-64\right)\times 36864}}{2\left(-64\right)}$$

Square -2304.

$$c=\frac{-\left(-2304\right)±\sqrt{5308416-4\left(-64\right)\times 36864}}{2\left(-64\right)}$$

Multiply -4 times -64.

$$c=\frac{-\left(-2304\right)±\sqrt{5308416+256\times 36864}}{2\left(-64\right)}$$

Multiply 256 times 36864.

$$c=\frac{-\left(-2304\right)±\sqrt{5308416+9437184}}{2\left(-64\right)}$$

Add 5308416 to 9437184.

$$c=\frac{-\left(-2304\right)±\sqrt{14745600}}{2\left(-64\right)}$$

Take the square root of 14745600.

$$c=\frac{-\left(-2304\right)±3840}{2\left(-64\right)}$$

The opposite of -2304 is 2304.

$$c=\frac{2304±3840}{2\left(-64\right)}$$

Multiply 2 times -64.

$$c=\frac{2304±3840}{-128}$$

Now solve the equation c=\frac{2304±3840}{-128} when ± is plus. Add 2304 to 3840.

$$c=\frac{6144}{-128}$$

Divide 6144 by -128.

$$c=-48$$

Now solve the equation c=\frac{2304±3840}{-128} when ± is minus. Subtract 3840 from 2304.

$$c=-\frac{1536}{-128}$$

Divide -1536 by -128.

$$c=12$$

The equation is now solved.

$$c=-48 c=12$$

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.

$$-64c^{2}-2304c+36864=0$$

Subtract 36864 from both sides of the equation.

$$-64c^{2}-2304c+36864-36864=-36864$$

Subtracting 36864 from itself leaves 0.

$$-64c^{2}-2304c=-36864$$

Divide both sides by -64.

$$\frac{-64c^{2}-2304c}{-64}=-\frac{36864}{-64}$$

Dividing by -64 undoes the multiplication by -64.

$$c^{2}+\left(-\frac{2304}{-64}\right)c=-\frac{36864}{-64}$$

Divide -2304 by -64.

$$c^{2}+36c=-\frac{36864}{-64}$$

Divide -36864 by -64.

$$c^{2}+36c=576$$

Divide 36, the coefficient of the x term, by 2 to get 18. Then add the square of 18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}+36c+18^{2}=576+18^{2}$$

Square 18.

$$c^{2}+36c+324=576+324$$

Add 576 to 324.

$$c^{2}+36c+324=900$$

Factor c^{2}+36c+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(c+18\right)^{2}=900$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c+18\right)^{2}}=\sqrt{900}$$

Simplify.

$$c+18=30 c+18=-30$$

Subtract 18 from both sides of the equation.

$$c=12 c=-48$$