Solve -16x^2+16x+32

$$-16 { x }^{ 2 } +16x+32$$

Factor

$$-16\left(x-2\right)\left(x+1\right)$$

Solution Steps

Factor out 16.

$$16\left(-x^{2}+x+2\right)$$

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

$$a+b=1 ab=-2=-2$$

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. The only such pair is the system solution.

$$a=2 b=-1$$

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

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

Factor out -x in the first and -1 in the second group.

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

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

$$\left(x-2\right)\left(-x-1\right)$$

Rewrite the complete factored expression.

$$16\left(x-2\right)\left(-x-1\right)$$

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.

$$-16x^{2}+16x+32=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-16\right)\times 32}}{2\left(-16\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-16\right)\times 32}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-16±\sqrt{256+64\times 32}}{2\left(-16\right)}$$

Multiply 64 times 32.

$$x=\frac{-16±\sqrt{256+2048}}{2\left(-16\right)}$$

Add 256 to 2048.

$$x=\frac{-16±\sqrt{2304}}{2\left(-16\right)}$$

Take the square root of 2304.

$$x=\frac{-16±48}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-16±48}{-32}$$

Now solve the equation x=\frac{-16±48}{-32} when ± is plus. Add -16 to 48.

$$x=\frac{32}{-32}$$

Divide 32 by -32.

$$x=-1$$

Now solve the equation x=\frac{-16±48}{-32} when ± is minus. Subtract 48 from -16.

$$x=-\frac{64}{-32}$$

Divide -64 by -32.

$$x=2$$

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

$$-16x^{2}+16x+32=-16\left(x-\left(-1\right)\right)\left(x-2\right)$$

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

$$-16x^{2}+16x+32=-16\left(x+1\right)\left(x-2\right)$$

Evaluate

$$-16\left(x-2\right)\left(x+1\right)$$