Solve -6d^2+18=12d

$$- 6 d ^ { 2 } + 18 = 12 d$$

Solve for d

$$d=-3$$

$$d=1$$

Steps Using Factoring By Grouping

Subtract 12d from both sides.

$$-6d^{2}+18-12d=0$$

Divide both sides by 6.

$$-d^{2}+3-2d=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-d^{2}-2d+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -d^{2}+ad+bd+3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-3=-3$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. The only such pair is the system solution.

$$a=1 b=-3$$

Rewrite -d^{2}-2d+3 as \left(-d^{2}+d\right)+\left(-3d+3\right).

$$\left(-d^{2}+d\right)+\left(-3d+3\right)$$

Factor out d in the first and 3 in the second group.

$$d\left(-d+1\right)+3\left(-d+1\right)$$

Factor out common term -d+1 by using distributive property.

$$\left(-d+1\right)\left(d+3\right)$$

To find equation solutions, solve -d+1=0 and d+3=0.

$$d=1 d=-3$$

Steps Using the Quadratic Formula

Subtract 12d from both sides.

$$-6d^{2}+18-12d=0$$

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.

$$-6d^{2}-12d+18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -12 for b, and 18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-6\right)\times 18}}{2\left(-6\right)}$$

Square -12.

$$d=\frac{-\left(-12\right)±\sqrt{144-4\left(-6\right)\times 18}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$d=\frac{-\left(-12\right)±\sqrt{144+24\times 18}}{2\left(-6\right)}$$

Multiply 24 times 18.

$$d=\frac{-\left(-12\right)±\sqrt{144+432}}{2\left(-6\right)}$$

Add 144 to 432.

$$d=\frac{-\left(-12\right)±\sqrt{576}}{2\left(-6\right)}$$

Take the square root of 576.

$$d=\frac{-\left(-12\right)±24}{2\left(-6\right)}$$

The opposite of -12 is 12.

$$d=\frac{12±24}{2\left(-6\right)}$$

Multiply 2 times -6.

$$d=\frac{12±24}{-12}$$

Now solve the equation d=\frac{12±24}{-12} when ± is plus. Add 12 to 24.

$$d=\frac{36}{-12}$$

Divide 36 by -12.

$$d=-3$$

Now solve the equation d=\frac{12±24}{-12} when ± is minus. Subtract 24 from 12.

$$d=-\frac{12}{-12}$$

Divide -12 by -12.

$$d=1$$

The equation is now solved.

$$d=-3 d=1$$

Steps for Completing the Square

Subtract 12d from both sides.

$$-6d^{2}+18-12d=0$$

Subtract 18 from both sides. Anything subtracted from zero gives its negation.

$$-6d^{2}-12d=-18$$

Divide both sides by -6.

$$\frac{-6d^{2}-12d}{-6}=-\frac{18}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$d^{2}+\left(-\frac{12}{-6}\right)d=-\frac{18}{-6}$$

Divide -12 by -6.

$$d^{2}+2d=-\frac{18}{-6}$$

Divide -18 by -6.

$$d^{2}+2d=3$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+2d+1^{2}=3+1^{2}$$

Square 1.

$$d^{2}+2d+1=3+1$$

Add 3 to 1.

$$d^{2}+2d+1=4$$

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

$$\left(d+1\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+1\right)^{2}}=\sqrt{4}$$

Simplify.

$$d+1=2 d+1=-2$$

Subtract 1 from both sides of the equation.

$$d=1 d=-3$$