Solve (36-6d)+36-6d/d+3d+3=36

$$( 36 - 6 d ) + \frac { 36 - 6 d } { d } + 3 d + 3 = 36$$

Solve for d

$$d=-4$$

$$d=3$$

Steps Using Factoring By Grouping

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by d.

$$d\times 36-6dd+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3d^{2}+d\times 3=36d$$

Combine d\times 36 and -6d to get 30d.

$$30d-6d^{2}+36+3d^{2}+d\times 3=36d$$

Combine -6d^{2} and 3d^{2} to get -3d^{2}.

$$30d-3d^{2}+36+d\times 3=36d$$

Combine 30d and d\times 3 to get 33d.

$$33d-3d^{2}+36=36d$$

Subtract 36d from both sides.

$$33d-3d^{2}+36-36d=0$$

Combine 33d and -36d to get -3d.

$$-3d-3d^{2}+36=0$$

Divide both sides by 3.

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

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

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -d^{2}+ad+bd+12. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-12=-12$$

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. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=3 b=-4$$

Rewrite -d^{2}-d+12 as \left(-d^{2}+3d\right)+\left(-4d+12\right).

$$\left(-d^{2}+3d\right)+\left(-4d+12\right)$$

Factor out d in the first and 4 in the second group.

$$d\left(-d+3\right)+4\left(-d+3\right)$$

Factor out common term -d+3 by using distributive property.

$$\left(-d+3\right)\left(d+4\right)$$

To find equation solutions, solve -d+3=0 and d+4=0.

$$d=3 d=-4$$

Steps Using the Quadratic Formula

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by d.

$$d\times 36-6dd+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3d^{2}+d\times 3=36d$$

Combine d\times 36 and -6d to get 30d.

$$30d-6d^{2}+36+3d^{2}+d\times 3=36d$$

Combine -6d^{2} and 3d^{2} to get -3d^{2}.

$$30d-3d^{2}+36+d\times 3=36d$$

Combine 30d and d\times 3 to get 33d.

$$33d-3d^{2}+36=36d$$

Subtract 36d from both sides.

$$33d-3d^{2}+36-36d=0$$

Combine 33d and -36d to get -3d.

$$-3d-3d^{2}+36=0$$

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.

$$-3d^{2}-3d+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -3 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-3\right)\times 36}}{2\left(-3\right)}$$

Square -3.

$$d=\frac{-\left(-3\right)±\sqrt{9-4\left(-3\right)\times 36}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$d=\frac{-\left(-3\right)±\sqrt{9+12\times 36}}{2\left(-3\right)}$$

Multiply 12 times 36.

$$d=\frac{-\left(-3\right)±\sqrt{9+432}}{2\left(-3\right)}$$

Add 9 to 432.

$$d=\frac{-\left(-3\right)±\sqrt{441}}{2\left(-3\right)}$$

Take the square root of 441.

$$d=\frac{-\left(-3\right)±21}{2\left(-3\right)}$$

The opposite of -3 is 3.

$$d=\frac{3±21}{2\left(-3\right)}$$

Multiply 2 times -3.

$$d=\frac{3±21}{-6}$$

Now solve the equation d=\frac{3±21}{-6} when ± is plus. Add 3 to 21.

$$d=\frac{24}{-6}$$

Divide 24 by -6.

$$d=-4$$

Now solve the equation d=\frac{3±21}{-6} when ± is minus. Subtract 21 from 3.

$$d=-\frac{18}{-6}$$

Divide -18 by -6.

$$d=3$$

The equation is now solved.

$$d=-4 d=3$$

Steps for Completing the Square

Variable d cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by d.

$$d\times 36-6dd+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3dd+d\times 3=36d$$

Multiply d and d to get d^{2}.

$$d\times 36-6d^{2}+36-6d+3d^{2}+d\times 3=36d$$

Combine d\times 36 and -6d to get 30d.

$$30d-6d^{2}+36+3d^{2}+d\times 3=36d$$

Combine -6d^{2} and 3d^{2} to get -3d^{2}.

$$30d-3d^{2}+36+d\times 3=36d$$

Combine 30d and d\times 3 to get 33d.

$$33d-3d^{2}+36=36d$$

Subtract 36d from both sides.

$$33d-3d^{2}+36-36d=0$$

Combine 33d and -36d to get -3d.

$$-3d-3d^{2}+36=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$-3d-3d^{2}=-36$$

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.

$$-3d^{2}-3d=-36$$

Divide both sides by -3.

$$\frac{-3d^{2}-3d}{-3}=-\frac{36}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$d^{2}+\left(-\frac{3}{-3}\right)d=-\frac{36}{-3}$$

Divide -3 by -3.

$$d^{2}+d=-\frac{36}{-3}$$

Divide -36 by -3.

$$d^{2}+d=12$$

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

$$d^{2}+d+\left(\frac{1}{2}\right)^{2}=12+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+d+\frac{1}{4}=12+\frac{1}{4}$$

Add 12 to \frac{1}{4}.

$$d^{2}+d+\frac{1}{4}=\frac{49}{4}$$

Factor d^{2}+d+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d+\frac{1}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{1}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$d+\frac{1}{2}=\frac{7}{2} d+\frac{1}{2}=-\frac{7}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$d=3 d=-4$$