Solve 3-2x^2-90x

$$3 - 2 x ^ { 2 } - 90 x$$

Factor

$$-2\left(x-\frac{-\sqrt{2031}-45}{2}\right)\left(x-\frac{\sqrt{2031}-45}{2}\right)$$

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.

$$-2x^{2}-90x+3=0$$

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.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\left(-2\right)\times 3}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-90\right)±\sqrt{8100+8\times 3}}{2\left(-2\right)}$$

Multiply 8 times 3.

$$x=\frac{-\left(-90\right)±\sqrt{8100+24}}{2\left(-2\right)}$$

Add 8100 to 24.

$$x=\frac{-\left(-90\right)±\sqrt{8124}}{2\left(-2\right)}$$

Take the square root of 8124.

$$x=\frac{-\left(-90\right)±2\sqrt{2031}}{2\left(-2\right)}$$

The opposite of -90 is 90.

$$x=\frac{90±2\sqrt{2031}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{90±2\sqrt{2031}}{-4}$$

Now solve the equation x=\frac{90±2\sqrt{2031}}{-4} when ± is plus. Add 90 to 2\sqrt{2031}.

$$x=\frac{2\sqrt{2031}+90}{-4}$$

Divide 90+2\sqrt{2031} by -4.

$$x=\frac{-\sqrt{2031}-45}{2}$$

Now solve the equation x=\frac{90±2\sqrt{2031}}{-4} when ± is minus. Subtract 2\sqrt{2031} from 90.

$$x=\frac{90-2\sqrt{2031}}{-4}$$

Divide 90-2\sqrt{2031} by -4.

$$x=\frac{\sqrt{2031}-45}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-45-\sqrt{2031}}{2} for x_{1} and \frac{-45+\sqrt{2031}}{2} for x_{2}.

$$-2x^{2}-90x+3=-2\left(x-\frac{-\sqrt{2031}-45}{2}\right)\left(x-\frac{\sqrt{2031}-45}{2}\right)$$

Evaluate

$$3-90x-2x^{2}$$