Solve F(x)=16-4x-x^2

$$F ( x ) = 16 - 4 x - x ^ { 2 }$$

Factor

$$-\left(x-\left(-2\sqrt{5}-2\right)\right)\left(x-\left(2\sqrt{5}-2\right)\right)$$

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.

$$-x^{2}-4x+16=0$$

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.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 16}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 16}}{2\left(-1\right)}$$

Multiply 4 times 16.

$$x=\frac{-\left(-4\right)±\sqrt{16+64}}{2\left(-1\right)}$$

Add 16 to 64.

$$x=\frac{-\left(-4\right)±\sqrt{80}}{2\left(-1\right)}$$

Take the square root of 80.

$$x=\frac{-\left(-4\right)±4\sqrt{5}}{2\left(-1\right)}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{4±4\sqrt{5}}{-2}$$

Now solve the equation x=\frac{4±4\sqrt{5}}{-2} when ± is plus. Add 4 to 4\sqrt{5}.

$$x=\frac{4\sqrt{5}+4}{-2}$$

Divide 4+4\sqrt{5} by -2.

$$x=-2\sqrt{5}-2$$

Now solve the equation x=\frac{4±4\sqrt{5}}{-2} when ± is minus. Subtract 4\sqrt{5} from 4.

$$x=\frac{4-4\sqrt{5}}{-2}$$

Divide 4-4\sqrt{5} by -2.

$$x=2\sqrt{5}-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -2-2\sqrt{5} for x_{1} and -2+2\sqrt{5} for x_{2}.

$$-x^{2}-4x+16=-\left(x-\left(-2\sqrt{5}-2\right)\right)\left(x-\left(2\sqrt{5}-2\right)\right)$$

Evaluate

$$16-4x-x^{2}$$