Solve f(x)=8-2x-(x^2)

$$f ( x ) = 8 - 2 x - ( x ^ { 2 } )$$

Factor

$$\left(2-x\right)\left(x+4\right)$$

Solution Steps

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

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

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8=-8$$

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 -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=2 b=-4$$

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

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

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

$$x\left(-x+2\right)+4\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(x+4\right)$$

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.

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

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.

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 8.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\left(-1\right)}$$

Add 4 to 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$x=\frac{-\left(-2\right)±6}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±6}{-2}$$

Now solve the equation x=\frac{2±6}{-2} when ± is plus. Add 2 to 6.

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

Divide 8 by -2.

$$x=-4$$

Now solve the equation x=\frac{2±6}{-2} when ± is minus. Subtract 6 from 2.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

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 2 for x_{2}.

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

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

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

Evaluate

$$\left(2-x\right)\left(x+4\right)$$