Solve -x^2+6x-2

$$- { x }^{ 2 } +6x-2$$

Factor

$$-\left(x-\left(3-\sqrt{7}\right)\right)\left(x-\left(\sqrt{7}+3\right)\right)$$

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.

$$-x^{2}+6x-2=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\left(-2\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\left(-2\right)}}{2\left(-1\right)}$$

Multiply 4 times -2.

$$x=\frac{-6±\sqrt{36-8}}{2\left(-1\right)}$$

Add 36 to -8.

$$x=\frac{-6±\sqrt{28}}{2\left(-1\right)}$$

Take the square root of 28.

$$x=\frac{-6±2\sqrt{7}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±2\sqrt{7}}{-2}$$

Now solve the equation x=\frac{-6±2\sqrt{7}}{-2} when ± is plus. Add -6 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-6}{-2}$$

Divide -6+2\sqrt{7} by -2.

$$x=3-\sqrt{7}$$

Now solve the equation x=\frac{-6±2\sqrt{7}}{-2} when ± is minus. Subtract 2\sqrt{7} from -6.

$$x=\frac{-2\sqrt{7}-6}{-2}$$

Divide -6-2\sqrt{7} by -2.

$$x=\sqrt{7}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3-\sqrt{7} for x_{1} and 3+\sqrt{7} for x_{2}.

$$-x^{2}+6x-2=-\left(x-\left(3-\sqrt{7}\right)\right)\left(x-\left(\sqrt{7}+3\right)\right)$$

Evaluate

$$-x^{2}+6x-2$$