Solve {l}{6x-2x^2}{+7}

$$\left. \begin{array} { l } { 6 x - 2 x ^ { 2 } } \\ { + 7 } \end{array} \right.$$

Factor

$$-2\left(x-\frac{3-\sqrt{23}}{2}\right)\left(x-\frac{\sqrt{23}+3}{2}\right)$$

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.

$$-2x^{2}+6x+7=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-2\right)\times 7}}{2\left(-2\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-2\right)\times 7}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-6±\sqrt{36+8\times 7}}{2\left(-2\right)}$$

Multiply 8 times 7.

$$x=\frac{-6±\sqrt{36+56}}{2\left(-2\right)}$$

Add 36 to 56.

$$x=\frac{-6±\sqrt{92}}{2\left(-2\right)}$$

Take the square root of 92.

$$x=\frac{-6±2\sqrt{23}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-6±2\sqrt{23}}{-4}$$

Now solve the equation x=\frac{-6±2\sqrt{23}}{-4} when ± is plus. Add -6 to 2\sqrt{23}.

$$x=\frac{2\sqrt{23}-6}{-4}$$

Divide -6+2\sqrt{23} by -4.

$$x=\frac{3-\sqrt{23}}{2}$$

Now solve the equation x=\frac{-6±2\sqrt{23}}{-4} when ± is minus. Subtract 2\sqrt{23} from -6.

$$x=\frac{-2\sqrt{23}-6}{-4}$$

Divide -6-2\sqrt{23} by -4.

$$x=\frac{\sqrt{23}+3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{23}}{2} for x_{1} and \frac{3+\sqrt{23}}{2} for x_{2}.

$$-2x^{2}+6x+7=-2\left(x-\frac{3-\sqrt{23}}{2}\right)\left(x-\frac{\sqrt{23}+3}{2}\right)$$

Evaluate

$$7+6x-2x^{2}$$