Solve -2x^2+8x-3

$$-2 { x }^{ 2 } +8x-3$$

Factor

$$-2\left(x-\left(-\frac{\sqrt{10}}{2}+2\right)\right)\left(x-\left(\frac{\sqrt{10}}{2}+2\right)\right)$$

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.

$$-2x^{2}+8x-3=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\left(-3\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-8±\sqrt{64+8\left(-3\right)}}{2\left(-2\right)}$$

Multiply 8 times -3.

$$x=\frac{-8±\sqrt{64-24}}{2\left(-2\right)}$$

Add 64 to -24.

$$x=\frac{-8±\sqrt{40}}{2\left(-2\right)}$$

Take the square root of 40.

$$x=\frac{-8±2\sqrt{10}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-8±2\sqrt{10}}{-4}$$

Now solve the equation x=\frac{-8±2\sqrt{10}}{-4} when ± is plus. Add -8 to 2\sqrt{10}.

$$x=\frac{2\sqrt{10}-8}{-4}$$

Divide -8+2\sqrt{10} by -4.

$$x=-\frac{\sqrt{10}}{2}+2$$

Now solve the equation x=\frac{-8±2\sqrt{10}}{-4} when ± is minus. Subtract 2\sqrt{10} from -8.

$$x=\frac{-2\sqrt{10}-8}{-4}$$

Divide -8-2\sqrt{10} by -4.

$$x=\frac{\sqrt{10}}{2}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2-\frac{\sqrt{10}}{2} for x_{1} and 2+\frac{\sqrt{10}}{2} for x_{2}.

$$-2x^{2}+8x-3=-2\left(x-\left(-\frac{\sqrt{10}}{2}+2\right)\right)\left(x-\left(\frac{\sqrt{10}}{2}+2\right)\right)$$

Evaluate

$$-2x^{2}+8x-3$$