Solve -2x^2-5x+1

$$-2 { x }^{ 2 } -5x+1$$

Factor

$$-2\left(x-\frac{-\sqrt{33}-5}{4}\right)\left(x-\frac{\sqrt{33}-5}{4}\right)$$

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.

$$-2x^{2}-5x+1=0$$

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.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-2\right)}}{2\left(-2\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-2\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-5\right)±\sqrt{25+8}}{2\left(-2\right)}$$

Add 25 to 8.

$$x=\frac{-\left(-5\right)±\sqrt{33}}{2\left(-2\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{33}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{5±\sqrt{33}}{-4}$$

Now solve the equation x=\frac{5±\sqrt{33}}{-4} when ± is plus. Add 5 to \sqrt{33}.

$$x=\frac{\sqrt{33}+5}{-4}$$

Divide 5+\sqrt{33} by -4.

$$x=\frac{-\sqrt{33}-5}{4}$$

Now solve the equation x=\frac{5±\sqrt{33}}{-4} when ± is minus. Subtract \sqrt{33} from 5.

$$x=\frac{5-\sqrt{33}}{-4}$$

Divide 5-\sqrt{33} by -4.

$$x=\frac{\sqrt{33}-5}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-5-\sqrt{33}}{4} for x_{1} and \frac{-5+\sqrt{33}}{4} for x_{2}.

$$-2x^{2}-5x+1=-2\left(x-\frac{-\sqrt{33}-5}{4}\right)\left(x-\frac{\sqrt{33}-5}{4}\right)$$

Evaluate

$$1-5x-2x^{2}$$