Solve -3x^2-18x+12=0

$$-3 { x }^{ 2 } -18x+12=0$$

Solve for x (complex solution)

$$x=\sqrt{13}-3\approx 0.605551275$$

$$x=-\left(\sqrt{13}+3\right)\approx -6.605551275$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}-18x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -18 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

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

Multiply 12 times 12.

$$x=\frac{-\left(-18\right)±\sqrt{324+144}}{2\left(-3\right)}$$

Add 324 to 144.

$$x=\frac{-\left(-18\right)±\sqrt{468}}{2\left(-3\right)}$$

Take the square root of 468.

$$x=\frac{-\left(-18\right)±6\sqrt{13}}{2\left(-3\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±6\sqrt{13}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{18±6\sqrt{13}}{-6}$$

Now solve the equation x=\frac{18±6\sqrt{13}}{-6} when ± is plus. Add 18 to 6\sqrt{13}.

$$x=\frac{6\sqrt{13}+18}{-6}$$

Divide 18+6\sqrt{13} by -6.

$$x=-\left(\sqrt{13}+3\right)$$

Now solve the equation x=\frac{18±6\sqrt{13}}{-6} when ± is minus. Subtract 6\sqrt{13} from 18.

$$x=\frac{18-6\sqrt{13}}{-6}$$

Divide 18-6\sqrt{13} by -6.

$$x=\sqrt{13}-3$$

The equation is now solved.

$$x=-\left(\sqrt{13}+3\right) x=\sqrt{13}-3$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-3x^{2}-18x+12=0$$

Subtract 12 from both sides of the equation.

$$-3x^{2}-18x+12-12=-12$$

Subtracting 12 from itself leaves 0.

$$-3x^{2}-18x=-12$$

Divide both sides by -3.

$$\frac{-3x^{2}-18x}{-3}=-\frac{12}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{18}{-3}\right)x=-\frac{12}{-3}$$

Divide -18 by -3.

$$x^{2}+6x=-\frac{12}{-3}$$

Divide -12 by -3.

$$x^{2}+6x=4$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=4+3^{2}$$

Square 3.

$$x^{2}+6x+9=4+9$$

Add 4 to 9.

$$x^{2}+6x+9=13$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{13}$$

Simplify.

$$x+3=\sqrt{13} x+3=-\sqrt{13}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{13}-3 x=-\sqrt{13}-3$$

Solve for x

$$x=\sqrt{13}-3\approx 0.605551275$$

$$x=-\sqrt{13}-3\approx -6.605551275$$

Steps Using the Quadratic Formula

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.

$$-3x^{2}-18x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -18 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\left(-3\right)\times 12}}{2\left(-3\right)}$$

Multiply -4 times -3.

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

Multiply 12 times 12.

$$x=\frac{-\left(-18\right)±\sqrt{324+144}}{2\left(-3\right)}$$

Add 324 to 144.

$$x=\frac{-\left(-18\right)±\sqrt{468}}{2\left(-3\right)}$$

Take the square root of 468.

$$x=\frac{-\left(-18\right)±6\sqrt{13}}{2\left(-3\right)}$$

The opposite of -18 is 18.

$$x=\frac{18±6\sqrt{13}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{18±6\sqrt{13}}{-6}$$

Now solve the equation x=\frac{18±6\sqrt{13}}{-6} when ± is plus. Add 18 to 6\sqrt{13}.

$$x=\frac{6\sqrt{13}+18}{-6}$$

Divide 18+6\sqrt{13} by -6.

$$x=-\left(\sqrt{13}+3\right)$$

Now solve the equation x=\frac{18±6\sqrt{13}}{-6} when ± is minus. Subtract 6\sqrt{13} from 18.

$$x=\frac{18-6\sqrt{13}}{-6}$$

Divide 18-6\sqrt{13} by -6.

$$x=\sqrt{13}-3$$

The equation is now solved.

$$x=-\left(\sqrt{13}+3\right) x=\sqrt{13}-3$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-3x^{2}-18x+12=0$$

Subtract 12 from both sides of the equation.

$$-3x^{2}-18x+12-12=-12$$

Subtracting 12 from itself leaves 0.

$$-3x^{2}-18x=-12$$

Divide both sides by -3.

$$\frac{-3x^{2}-18x}{-3}=-\frac{12}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{18}{-3}\right)x=-\frac{12}{-3}$$

Divide -18 by -3.

$$x^{2}+6x=-\frac{12}{-3}$$

Divide -12 by -3.

$$x^{2}+6x=4$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=4+3^{2}$$

Square 3.

$$x^{2}+6x+9=4+9$$

Add 4 to 9.

$$x^{2}+6x+9=13$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=13$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{13}$$

Simplify.

$$x+3=\sqrt{13} x+3=-\sqrt{13}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{13}-3 x=-\sqrt{13}-3$$