Solve 8x-x^2-14

$$8x- { x }^{ 2 } -14$$

Factor

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

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.

$$-x^{2}+8x-14=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-1\right)\left(-14\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-8±\sqrt{64+4\left(-14\right)}}{2\left(-1\right)}$$

Multiply 4 times -14.

$$x=\frac{-8±\sqrt{64-56}}{2\left(-1\right)}$$

Add 64 to -56.

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

Take the square root of 8.

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

Multiply 2 times -1.

$$x=\frac{-8±2\sqrt{2}}{-2}$$

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

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

Divide 2\sqrt{2}-8 by -2.

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

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

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

Divide -8-2\sqrt{2} by -2.

$$x=\sqrt{2}+4$$

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

$$-x^{2}+8x-14=-\left(x-\left(4-\sqrt{2}\right)\right)\left(x-\left(\sqrt{2}+4\right)\right)$$

Evaluate

$$-x^{2}+8x-14$$