Solve 3-2x^2+18x-36

$$3 - 2 x ^ { 2 } + 18 x - 36$$

Factor

$$-2\left(x-\frac{9-\sqrt{15}}{2}\right)\left(x-\frac{\sqrt{15}+9}{2}\right)$$

Steps Using the Quadratic Formula

Subtract 36 from 3 to get -33.

$$factor(-33-2x^{2}+18x)$$

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.

$$-2x^{2}+18x-33=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-2\right)\left(-33\right)}}{2\left(-2\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-2\right)\left(-33\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-18±\sqrt{324+8\left(-33\right)}}{2\left(-2\right)}$$

Multiply 8 times -33.

$$x=\frac{-18±\sqrt{324-264}}{2\left(-2\right)}$$

Add 324 to -264.

$$x=\frac{-18±\sqrt{60}}{2\left(-2\right)}$$

Take the square root of 60.

$$x=\frac{-18±2\sqrt{15}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-18±2\sqrt{15}}{-4}$$

Now solve the equation x=\frac{-18±2\sqrt{15}}{-4} when ± is plus. Add -18 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-18}{-4}$$

Divide -18+2\sqrt{15} by -4.

$$x=\frac{9-\sqrt{15}}{2}$$

Now solve the equation x=\frac{-18±2\sqrt{15}}{-4} when ± is minus. Subtract 2\sqrt{15} from -18.

$$x=\frac{-2\sqrt{15}-18}{-4}$$

Divide -18-2\sqrt{15} by -4.

$$x=\frac{\sqrt{15}+9}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9-\sqrt{15}}{2} for x_{1} and \frac{9+\sqrt{15}}{2} for x_{2}.

$$-2x^{2}+18x-33=-2\left(x-\frac{9-\sqrt{15}}{2}\right)\left(x-\frac{\sqrt{15}+9}{2}\right)$$

Evaluate

$$-2x^{2}+18x-33$$

Solution Steps

Subtract 36 from 3 to get -33.

$$-33-2x^{2}+18x$$