Solve (2x+8)(x+4)+9x^2-4=0

$$( 2 x + 8 ) ( x + 4 ) + 9 x ^ { 2 } - 4 = 0$$

Solve for x (complex solution)

$$x=\frac{-8+2\sqrt{61}i}{11}\approx -0.727272727+1.420045396i$$

$$x=\frac{-2\sqrt{61}i-8}{11}\approx -0.727272727-1.420045396i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+8 by x+4 and combine like terms.

$$2x^{2}+16x+32+9x^{2}-4=0$$

Combine 2x^{2} and 9x^{2} to get 11x^{2}.

$$11x^{2}+16x+32-4=0$$

Subtract 4 from 32 to get 28.

$$11x^{2}+16x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 11 for a, 16 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 11\times 28}}{2\times 11}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 11\times 28}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-16±\sqrt{256-44\times 28}}{2\times 11}$$

Multiply -44 times 28.

$$x=\frac{-16±\sqrt{256-1232}}{2\times 11}$$

Add 256 to -1232.

$$x=\frac{-16±\sqrt{-976}}{2\times 11}$$

Take the square root of -976.

$$x=\frac{-16±4\sqrt{61}i}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{-16±4\sqrt{61}i}{22}$$

Now solve the equation x=\frac{-16±4\sqrt{61}i}{22} when ± is plus. Add -16 to 4i\sqrt{61}.

$$x=\frac{-16+4\sqrt{61}i}{22}$$

Divide -16+4i\sqrt{61} by 22.

$$x=\frac{-8+2\sqrt{61}i}{11}$$

Now solve the equation x=\frac{-16±4\sqrt{61}i}{22} when ± is minus. Subtract 4i\sqrt{61} from -16.

$$x=\frac{-4\sqrt{61}i-16}{22}$$

Divide -16-4i\sqrt{61} by 22.

$$x=\frac{-2\sqrt{61}i-8}{11}$$

The equation is now solved.

$$x=\frac{-8+2\sqrt{61}i}{11} x=\frac{-2\sqrt{61}i-8}{11}$$

Steps for Completing the Square

Use the distributive property to multiply 2x+8 by x+4 and combine like terms.

$$2x^{2}+16x+32+9x^{2}-4=0$$

Combine 2x^{2} and 9x^{2} to get 11x^{2}.

$$11x^{2}+16x+32-4=0$$

Subtract 4 from 32 to get 28.

$$11x^{2}+16x+28=0$$

Subtract 28 from both sides. Anything subtracted from zero gives its negation.

$$11x^{2}+16x=-28$$

Divide both sides by 11.

$$\frac{11x^{2}+16x}{11}=-\frac{28}{11}$$

Dividing by 11 undoes the multiplication by 11.

$$x^{2}+\frac{16}{11}x=-\frac{28}{11}$$

Divide \frac{16}{11}, the coefficient of the x term, by 2 to get \frac{8}{11}. Then add the square of \frac{8}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{11}x+\left(\frac{8}{11}\right)^{2}=-\frac{28}{11}+\left(\frac{8}{11}\right)^{2}$$

Square \frac{8}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{11}x+\frac{64}{121}=-\frac{28}{11}+\frac{64}{121}$$

Add -\frac{28}{11} to \frac{64}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{11}x+\frac{64}{121}=-\frac{244}{121}$$

Factor x^{2}+\frac{16}{11}x+\frac{64}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{11}\right)^{2}=-\frac{244}{121}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{11}\right)^{2}}=\sqrt{-\frac{244}{121}}$$

Simplify.

$$x+\frac{8}{11}=\frac{2\sqrt{61}i}{11} x+\frac{8}{11}=-\frac{2\sqrt{61}i}{11}$$

Subtract \frac{8}{11} from both sides of the equation.

$$x=\frac{-8+2\sqrt{61}i}{11} x=\frac{-2\sqrt{61}i-8}{11}$$