Solve f(x)=16-x^2+2x

$$f ( x ) = 16 - x ^ { 2 } + 2 x$$

Factor

$$-\left(x-\left(1-\sqrt{17}\right)\right)\left(x-\left(\sqrt{17}+1\right)\right)$$

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.

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

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.

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

Square 2.

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

Multiply -4 times -1.

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

Multiply 4 times 16.

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

Add 4 to 64.

$$x=\frac{-2±\sqrt{68}}{2\left(-1\right)}$$

Take the square root of 68.

$$x=\frac{-2±2\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-2±2\sqrt{17}}{-2}$$

Now solve the equation x=\frac{-2±2\sqrt{17}}{-2} when ± is plus. Add -2 to 2\sqrt{17}.

$$x=\frac{2\sqrt{17}-2}{-2}$$

Divide -2+2\sqrt{17} by -2.

$$x=1-\sqrt{17}$$

Now solve the equation x=\frac{-2±2\sqrt{17}}{-2} when ± is minus. Subtract 2\sqrt{17} from -2.

$$x=\frac{-2\sqrt{17}-2}{-2}$$

Divide -2-2\sqrt{17} by -2.

$$x=\sqrt{17}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1-\sqrt{17} for x_{1} and 1+\sqrt{17} for x_{2}.

$$-x^{2}+2x+16=-\left(x-\left(1-\sqrt{17}\right)\right)\left(x-\left(\sqrt{17}+1\right)\right)$$

Evaluate

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