Solve 48+8x-x^2

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

Factor

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

Solution Steps

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

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

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+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 positive, the positive number has greater absolute value than the negative. 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.

$$a=12 b=-4$$

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

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

Factor out -x in the first and -4 in the second group.

$$-x\left(x-12\right)-4\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(-x-4\right)$$

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.

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

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.

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

Square 8.

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

Multiply -4 times -1.

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

Multiply 4 times 48.

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

Add 64 to 192.

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

Take the square root of 256.

$$x=\frac{-8±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-8±16}{-2}$$

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

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

Divide 8 by -2.

$$x=-4$$

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

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

Divide -24 by -2.

$$x=12$$

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

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

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

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

Evaluate

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