Solve 3sqrt{3}+3d=12sqrt{3}+66d^2

$$3 \sqrt { 3 } + 3 d = 12 \sqrt { 3 } + 66 d ^ { 2 }$$

Solve for d

$$d=\frac{-i\sqrt{264\sqrt{3}-1}+1}{44}\approx 0.022727273-0.48546083i$$

$$d=\frac{1+i\sqrt{264\sqrt{3}-1}}{44}\approx 0.022727273+0.48546083i$$

Steps Using the Quadratic Formula

Subtract 12\sqrt{3} from both sides.

$$3\sqrt{3}+3d-12\sqrt{3}=66d^{2}$$

Combine 3\sqrt{3} and -12\sqrt{3} to get -9\sqrt{3}.

$$-9\sqrt{3}+3d=66d^{2}$$

Subtract 66d^{2} from both sides.

$$-9\sqrt{3}+3d-66d^{2}=0$$

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.

$$-66d^{2}+3d-9\sqrt{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -66 for a, 3 for b, and -9\sqrt{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-3±\sqrt{3^{2}-4\left(-66\right)\left(-9\sqrt{3}\right)}}{2\left(-66\right)}$$

Square 3.

$$d=\frac{-3±\sqrt{9-4\left(-66\right)\left(-9\sqrt{3}\right)}}{2\left(-66\right)}$$

Multiply -4 times -66.

$$d=\frac{-3±\sqrt{9+264\left(-9\sqrt{3}\right)}}{2\left(-66\right)}$$

Multiply 264 times -9\sqrt{3}.

$$d=\frac{-3±\sqrt{9-2376\sqrt{3}}}{2\left(-66\right)}$$

Take the square root of 9-2376\sqrt{3}.

$$d=\frac{-3±3i\sqrt{-\left(1-264\sqrt{3}\right)}}{2\left(-66\right)}$$

Multiply 2 times -66.

$$d=\frac{-3±3i\sqrt{-\left(1-264\sqrt{3}\right)}}{-132}$$

Now solve the equation d=\frac{-3±3i\sqrt{-\left(1-264\sqrt{3}\right)}}{-132} when ± is plus. Add -3 to 3i\sqrt{-\left(1-264\sqrt{3}\right)}.

$$d=\frac{-3+3i\sqrt{264\sqrt{3}-1}}{-132}$$

Divide -3+3i\sqrt{-1+264\sqrt{3}} by -132.

$$d=\frac{-i\sqrt{264\sqrt{3}-1}+1}{44}$$

Now solve the equation d=\frac{-3±3i\sqrt{-\left(1-264\sqrt{3}\right)}}{-132} when ± is minus. Subtract 3i\sqrt{-\left(1-264\sqrt{3}\right)} from -3.

$$d=\frac{-3i\sqrt{264\sqrt{3}-1}-3}{-132}$$

Divide -3-3i\sqrt{-1+264\sqrt{3}} by -132.

$$d=\frac{1+i\sqrt{264\sqrt{3}-1}}{44}$$

The equation is now solved.

$$d=\frac{-i\sqrt{264\sqrt{3}-1}+1}{44} d=\frac{1+i\sqrt{264\sqrt{3}-1}}{44}$$

Steps for Completing the Square

Subtract 66d^{2} from both sides.

$$3\sqrt{3}+3d-66d^{2}=12\sqrt{3}$$

Subtract 3\sqrt{3} from both sides.

$$3d-66d^{2}=12\sqrt{3}-3\sqrt{3}$$

Combine 12\sqrt{3} and -3\sqrt{3} to get 9\sqrt{3}.

$$3d-66d^{2}=9\sqrt{3}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-66d^{2}+3d=9\sqrt{3}$$

Divide both sides by -66.

$$\frac{-66d^{2}+3d}{-66}=\frac{3^{\frac{5}{2}}}{-66}$$

Dividing by -66 undoes the multiplication by -66.

$$d^{2}+\frac{3}{-66}d=\frac{3^{\frac{5}{2}}}{-66}$$

Reduce the fraction \frac{3}{-66} to lowest terms by extracting and canceling out 3.

$$d^{2}-\frac{1}{22}d=\frac{3^{\frac{5}{2}}}{-66}$$

Divide 3^{\frac{5}{2}} by -66.

$$d^{2}-\frac{1}{22}d=-\frac{3\sqrt{3}}{22}$$

Divide -\frac{1}{22}, the coefficient of the x term, by 2 to get -\frac{1}{44}. Then add the square of -\frac{1}{44} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{1}{22}d+\left(-\frac{1}{44}\right)^{2}=-\frac{3\sqrt{3}}{22}+\left(-\frac{1}{44}\right)^{2}$$

Square -\frac{1}{44} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{1}{22}d+\frac{1}{1936}=-\frac{3\sqrt{3}}{22}+\frac{1}{1936}$$

Factor d^{2}-\frac{1}{22}d+\frac{1}{1936}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{1}{44}\right)^{2}=-\frac{3\sqrt{3}}{22}+\frac{1}{1936}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{1}{44}\right)^{2}}=\sqrt{-\frac{3\sqrt{3}}{22}+\frac{1}{1936}}$$

Simplify.

$$d-\frac{1}{44}=\frac{i\sqrt{-\left(1-264\sqrt{3}\right)}}{44} d-\frac{1}{44}=-\frac{i\sqrt{264\sqrt{3}-1}}{44}$$

Add \frac{1}{44} to both sides of the equation.

$$d=\frac{1+i\sqrt{264\sqrt{3}-1}}{44} d=\frac{-i\sqrt{264\sqrt{3}-1}+1}{44}$$