Solve 4x(x+6)=-40

$$4 x ( x + 6 ) = - 40$$

Solve for x (complex solution)

$$x=-3+i$$

$$x=-3-i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+6.

$$4x^{2}+24x=-40$$

Add 40 to both sides.

$$4x^{2}+24x+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 24 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-24±\sqrt{24^{2}-4\times 4\times 40}}{2\times 4}$$

Square 24.

$$x=\frac{-24±\sqrt{576-4\times 4\times 40}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-24±\sqrt{576-16\times 40}}{2\times 4}$$

Multiply -16 times 40.

$$x=\frac{-24±\sqrt{576-640}}{2\times 4}$$

Add 576 to -640.

$$x=\frac{-24±\sqrt{-64}}{2\times 4}$$

Take the square root of -64.

$$x=\frac{-24±8i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-24±8i}{8}$$

Now solve the equation x=\frac{-24±8i}{8} when ± is plus. Add -24 to 8i.

$$x=\frac{-24+8i}{8}$$

Divide -24+8i by 8.

$$x=-3+i$$

Now solve the equation x=\frac{-24±8i}{8} when ± is minus. Subtract 8i from -24.

$$x=\frac{-24-8i}{8}$$

Divide -24-8i by 8.

$$x=-3-i$$

The equation is now solved.

$$x=-3+i x=-3-i$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+6.

$$4x^{2}+24x=-40$$

Divide both sides by 4.

$$\frac{4x^{2}+24x}{4}=-\frac{40}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{24}{4}x=-\frac{40}{4}$$

Divide 24 by 4.

$$x^{2}+6x=-\frac{40}{4}$$

Divide -40 by 4.

$$x^{2}+6x=-10$$

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.

$$x^{2}+6x+3^{2}=-10+3^{2}$$

Square 3.

$$x^{2}+6x+9=-10+9$$

Add -10 to 9.

$$x^{2}+6x+9=-1$$

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}.

$$\left(x+3\right)^{2}=-1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{-1}$$

Simplify.

$$x+3=i x+3=-i$$

Subtract 3 from both sides of the equation.

$$x=-3+i x=-3-i$$