Solve (1^3+6D^2+11D+6)4=0

$$( 1 ^ { 3 } + 6 D ^ { 2 } + 11 D + 6 ) 4 = 0$$

Solve for D

$$D=\frac{-11+\sqrt{47}i}{12}\approx -0.916666667+0.57130455i$$

$$D=\frac{-\sqrt{47}i-11}{12}\approx -0.916666667-0.57130455i$$

Steps Using the Quadratic Formula

Divide both sides by 4. Zero divided by any non-zero number gives zero.

$$1^{3}+6D^{2}+11D+6=0$$

Calculate 1 to the power of 3 and get 1.

$$1+6D^{2}+11D+6=0$$

Add 1 and 6 to get 7.

$$7+6D^{2}+11D=0$$

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.

$$6D^{2}+11D+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 11 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$D=\frac{-11±\sqrt{11^{2}-4\times 6\times 7}}{2\times 6}$$

Square 11.

$$D=\frac{-11±\sqrt{121-4\times 6\times 7}}{2\times 6}$$

Multiply -4 times 6.

$$D=\frac{-11±\sqrt{121-24\times 7}}{2\times 6}$$

Multiply -24 times 7.

$$D=\frac{-11±\sqrt{121-168}}{2\times 6}$$

Add 121 to -168.

$$D=\frac{-11±\sqrt{-47}}{2\times 6}$$

Take the square root of -47.

$$D=\frac{-11±\sqrt{47}i}{2\times 6}$$

Multiply 2 times 6.

$$D=\frac{-11±\sqrt{47}i}{12}$$

Now solve the equation D=\frac{-11±\sqrt{47}i}{12} when ± is plus. Add -11 to i\sqrt{47}.

$$D=\frac{-11+\sqrt{47}i}{12}$$

Now solve the equation D=\frac{-11±\sqrt{47}i}{12} when ± is minus. Subtract i\sqrt{47} from -11.

$$D=\frac{-\sqrt{47}i-11}{12}$$

The equation is now solved.

$$D=\frac{-11+\sqrt{47}i}{12} D=\frac{-\sqrt{47}i-11}{12}$$

Steps for Completing the Square

Divide both sides by 4. Zero divided by any non-zero number gives zero.

$$1^{3}+6D^{2}+11D+6=0$$

Calculate 1 to the power of 3 and get 1.

$$1+6D^{2}+11D+6=0$$

Add 1 and 6 to get 7.

$$7+6D^{2}+11D=0$$

Subtract 7 from both sides. Anything subtracted from zero gives its negation.

$$6D^{2}+11D=-7$$

Divide both sides by 6.

$$\frac{6D^{2}+11D}{6}=-\frac{7}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$D^{2}+\frac{11}{6}D=-\frac{7}{6}$$

Divide \frac{11}{6}, the coefficient of the x term, by 2 to get \frac{11}{12}. Then add the square of \frac{11}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$D^{2}+\frac{11}{6}D+\left(\frac{11}{12}\right)^{2}=-\frac{7}{6}+\left(\frac{11}{12}\right)^{2}$$

Square \frac{11}{12} by squaring both the numerator and the denominator of the fraction.

$$D^{2}+\frac{11}{6}D+\frac{121}{144}=-\frac{7}{6}+\frac{121}{144}$$

Add -\frac{7}{6} to \frac{121}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$D^{2}+\frac{11}{6}D+\frac{121}{144}=-\frac{47}{144}$$

Factor D^{2}+\frac{11}{6}D+\frac{121}{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}.

$$\left(D+\frac{11}{12}\right)^{2}=-\frac{47}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(D+\frac{11}{12}\right)^{2}}=\sqrt{-\frac{47}{144}}$$

Simplify.

$$D+\frac{11}{12}=\frac{\sqrt{47}i}{12} D+\frac{11}{12}=-\frac{\sqrt{47}i}{12}$$

Subtract \frac{11}{12} from both sides of the equation.

$$D=\frac{-11+\sqrt{47}i}{12} D=\frac{-\sqrt{47}i-11}{12}$$