Solve 4+3x-2x^2

$$4 + 3 x - 2 x ^ { 2 }$$

Factor

$$-2\left(x-\frac{3-\sqrt{41}}{4}\right)\left(x-\frac{\sqrt{41}+3}{4}\right)$$

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.

$$-2x^{2}+3x+4=0$$

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.

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

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-2\right)\times 4}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-3±\sqrt{9+8\times 4}}{2\left(-2\right)}$$

Multiply 8 times 4.

$$x=\frac{-3±\sqrt{9+32}}{2\left(-2\right)}$$

Add 9 to 32.

$$x=\frac{-3±\sqrt{41}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-3±\sqrt{41}}{-4}$$

Now solve the equation x=\frac{-3±\sqrt{41}}{-4} when ± is plus. Add -3 to \sqrt{41}.

$$x=\frac{\sqrt{41}-3}{-4}$$

Divide -3+\sqrt{41} by -4.

$$x=\frac{3-\sqrt{41}}{4}$$

Now solve the equation x=\frac{-3±\sqrt{41}}{-4} when ± is minus. Subtract \sqrt{41} from -3.

$$x=\frac{-\sqrt{41}-3}{-4}$$

Divide -3-\sqrt{41} by -4.

$$x=\frac{\sqrt{41}+3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3-\sqrt{41}}{4} for x_{1} and \frac{3+\sqrt{41}}{4} for x_{2}.

$$-2x^{2}+3x+4=-2\left(x-\frac{3-\sqrt{41}}{4}\right)\left(x-\frac{\sqrt{41}+3}{4}\right)$$

Evaluate

$$4+3x-2x^{2}$$