Solve 3x^2=-12x-15

$$3 x ^ { 2 } = - 12 x - 15$$

Solve for x (complex solution)

$$x=-2+i$$

$$x=-2-i$$

Steps Using the Quadratic Formula

Add 12x to both sides.

$$3x^{2}+12x=-15$$

Add 15 to both sides.

$$3x^{2}+12x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 12 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 3\times 15}}{2\times 3}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 3\times 15}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-12±\sqrt{144-12\times 15}}{2\times 3}$$

Multiply -12 times 15.

$$x=\frac{-12±\sqrt{144-180}}{2\times 3}$$

Add 144 to -180.

$$x=\frac{-12±\sqrt{-36}}{2\times 3}$$

Take the square root of -36.

$$x=\frac{-12±6i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-12±6i}{6}$$

Now solve the equation x=\frac{-12±6i}{6} when ± is plus. Add -12 to 6i.

$$x=\frac{-12+6i}{6}$$

Divide -12+6i by 6.

$$x=-2+i$$

Now solve the equation x=\frac{-12±6i}{6} when ± is minus. Subtract 6i from -12.

$$x=\frac{-12-6i}{6}$$

Divide -12-6i by 6.

$$x=-2-i$$

The equation is now solved.

$$x=-2+i x=-2-i$$

Steps for Completing the Square

Add 12x to both sides.

$$3x^{2}+12x=-15$$

Divide both sides by 3.

$$\frac{3x^{2}+12x}{3}=-\frac{15}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{12}{3}x=-\frac{15}{3}$$

Divide 12 by 3.

$$x^{2}+4x=-\frac{15}{3}$$

Divide -15 by 3.

$$x^{2}+4x=-5$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=-5+2^{2}$$

Square 2.

$$x^{2}+4x+4=-5+4$$

Add -5 to 4.

$$x^{2}+4x+4=-1$$

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+2\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x+2=i x+2=-i$$

Subtract 2 from both sides of the equation.

$$x=-2+i x=-2-i$$