Solve 1-2x^2+7x=

$$1 - 2 x ^ { 2 } + 7 x =$$

Factor

$$-2\left(x-\frac{7-\sqrt{57}}{4}\right)\left(x-\frac{\sqrt{57}+7}{4}\right)$$

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.

$$-2x^{2}+7x+1=0$$

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.

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

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-7±\sqrt{49+8}}{2\left(-2\right)}$$

Add 49 to 8.

$$x=\frac{-7±\sqrt{57}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-7±\sqrt{57}}{-4}$$

Now solve the equation x=\frac{-7±\sqrt{57}}{-4} when ± is plus. Add -7 to \sqrt{57}.

$$x=\frac{\sqrt{57}-7}{-4}$$

Divide -7+\sqrt{57} by -4.

$$x=\frac{7-\sqrt{57}}{4}$$

Now solve the equation x=\frac{-7±\sqrt{57}}{-4} when ± is minus. Subtract \sqrt{57} from -7.

$$x=\frac{-\sqrt{57}-7}{-4}$$

Divide -7-\sqrt{57} by -4.

$$x=\frac{\sqrt{57}+7}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7-\sqrt{57}}{4} for x_{1} and \frac{7+\sqrt{57}}{4} for x_{2}.

$$-2x^{2}+7x+1=-2\left(x-\frac{7-\sqrt{57}}{4}\right)\left(x-\frac{\sqrt{57}+7}{4}\right)$$

Evaluate

$$1+7x-2x^{2}$$