Solve -3x^2-18x-23

$$-3 { x }^{ 2 } -18x-23$$

Factor

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

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.

$$-3x^{2}-18x-23=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-3\right)\left(-23\right)}}{2\left(-3\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-3\right)\left(-23\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-18\right)±\sqrt{324+12\left(-23\right)}}{2\left(-3\right)}$$

Multiply 12 times -23.

$$x=\frac{-\left(-18\right)±\sqrt{324-276}}{2\left(-3\right)}$$

Add 324 to -276.

$$x=\frac{-\left(-18\right)±\sqrt{48}}{2\left(-3\right)}$$

Take the square root of 48.

$$x=\frac{-\left(-18\right)±4\sqrt{3}}{2\left(-3\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±4\sqrt{3}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{18±4\sqrt{3}}{-6}$$

Now solve the equation x=\frac{18±4\sqrt{3}}{-6} when ± is plus. Add 18 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+18}{-6}$$

Divide 18+4\sqrt{3} by -6.

$$x=-\frac{2\sqrt{3}}{3}-3$$

Now solve the equation x=\frac{18±4\sqrt{3}}{-6} when ± is minus. Subtract 4\sqrt{3} from 18.

$$x=\frac{18-4\sqrt{3}}{-6}$$

Divide 18-4\sqrt{3} by -6.

$$x=\frac{2\sqrt{3}}{3}-3$$

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

$$-3x^{2}-18x-23=-3\left(x-\left(-\frac{2\sqrt{3}}{3}-3\right)\right)\left(x-\left(\frac{2\sqrt{3}}{3}-3\right)\right)$$

Evaluate

$$-3x^{2}-18x-23$$