Solve left(2w+18right)left(2w+30right)+576=0

$$\left( 2w+18 \right) \left( 2w+30 \right) +576 = 0$$

Solve for w

$$w=-12+3\sqrt{15}i\approx -12+11.618950039i$$

$$w=-3\sqrt{15}i-12\approx -12-11.618950039i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2w+18 by 2w+30 and combine like terms.

$$4w^{2}+96w+540+576=0$$

Add 540 and 576 to get 1116.

$$4w^{2}+96w+1116=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 96 for b, and 1116 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-96±\sqrt{96^{2}-4\times 4\times 1116}}{2\times 4}$$

Square 96.

$$w=\frac{-96±\sqrt{9216-4\times 4\times 1116}}{2\times 4}$$

Multiply -4 times 4.

$$w=\frac{-96±\sqrt{9216-16\times 1116}}{2\times 4}$$

Multiply -16 times 1116.

$$w=\frac{-96±\sqrt{9216-17856}}{2\times 4}$$

Add 9216 to -17856.

$$w=\frac{-96±\sqrt{-8640}}{2\times 4}$$

Take the square root of -8640.

$$w=\frac{-96±24\sqrt{15}i}{2\times 4}$$

Multiply 2 times 4.

$$w=\frac{-96±24\sqrt{15}i}{8}$$

Now solve the equation w=\frac{-96±24\sqrt{15}i}{8} when ± is plus. Add -96 to 24i\sqrt{15}.

$$w=\frac{-96+24\sqrt{15}i}{8}$$

Divide -96+24i\sqrt{15} by 8.

$$w=-12+3\sqrt{15}i$$

Now solve the equation w=\frac{-96±24\sqrt{15}i}{8} when ± is minus. Subtract 24i\sqrt{15} from -96.

$$w=\frac{-24\sqrt{15}i-96}{8}$$

Divide -96-24i\sqrt{15} by 8.

$$w=-3\sqrt{15}i-12$$

The equation is now solved.

$$w=-12+3\sqrt{15}i w=-3\sqrt{15}i-12$$

Steps for Completing the Square

Use the distributive property to multiply 2w+18 by 2w+30 and combine like terms.

$$4w^{2}+96w+540+576=0$$

Add 540 and 576 to get 1116.

$$4w^{2}+96w+1116=0$$

Subtract 1116 from both sides. Anything subtracted from zero gives its negation.

$$4w^{2}+96w=-1116$$

Divide both sides by 4.

$$\frac{4w^{2}+96w}{4}=-\frac{1116}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$w^{2}+\frac{96}{4}w=-\frac{1116}{4}$$

Divide 96 by 4.

$$w^{2}+24w=-\frac{1116}{4}$$

Divide -1116 by 4.

$$w^{2}+24w=-279$$

Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+24w+12^{2}=-279+12^{2}$$

Square 12.

$$w^{2}+24w+144=-279+144$$

Add -279 to 144.

$$w^{2}+24w+144=-135$$

Factor w^{2}+24w+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+12\right)^{2}=-135$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+12\right)^{2}}=\sqrt{-135}$$

Simplify.

$$w+12=3\sqrt{15}i w+12=-3\sqrt{15}i$$

Subtract 12 from both sides of the equation.

$$w=-12+3\sqrt{15}i w=-3\sqrt{15}i-12$$