Solve 48-8c-c^2

$$48 - 8 c - c ^ { 2 }$$

Factor

$$\left(4-c\right)\left(c+12\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-c^{2}-8c+48$$

Factor the expression by grouping. First, the expression needs to be rewritten as -c^{2}+ac+bc+48. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-48=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -8.

$$a=4 b=-12$$

Rewrite -c^{2}-8c+48 as \left(-c^{2}+4c\right)+\left(-12c+48\right).

$$\left(-c^{2}+4c\right)+\left(-12c+48\right)$$

Factor out c in the first and 12 in the second group.

$$c\left(-c+4\right)+12\left(-c+4\right)$$

Factor out common term -c+4 by using distributive property.

$$\left(-c+4\right)\left(c+12\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-c^{2}-8c+48=0$$

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.

$$c=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Square -8.

$$c=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 48}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$c=\frac{-\left(-8\right)±\sqrt{64+4\times 48}}{2\left(-1\right)}$$

Multiply 4 times 48.

$$c=\frac{-\left(-8\right)±\sqrt{64+192}}{2\left(-1\right)}$$

Add 64 to 192.

$$c=\frac{-\left(-8\right)±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

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

The opposite of -8 is 8.

$$c=\frac{8±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$c=\frac{8±16}{-2}$$

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

$$c=\frac{24}{-2}$$

Divide 24 by -2.

$$c=-12$$

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

$$c=-\frac{8}{-2}$$

Divide -8 by -2.

$$c=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -12 for x_{1} and 4 for x_{2}.

$$-c^{2}-8c+48=-\left(c-\left(-12\right)\right)\left(c-4\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-c^{2}-8c+48=-\left(c+12\right)\left(c-4\right)$$

Evaluate

$$\left(4-c\right)\left(c+12\right)$$