Solve -6/4-d=2d/d-2

$$\frac { - 6 } { 4 - d } = \frac { 2 d } { d - 2 }$$

Solve for d

$$d=1$$

$$d=6$$

Steps Using Factoring By Grouping

Variable d cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by \left(d-4\right)\left(d-2\right), the least common multiple of 4-d,d-2.

$$\left(2-d\right)\left(-6\right)=\left(d-4\right)\times 2d$$

Use the distributive property to multiply 2-d by -6.

$$-12+6d=\left(d-4\right)\times 2d$$

Use the distributive property to multiply d-4 by 2.

$$-12+6d=\left(2d-8\right)d$$

Use the distributive property to multiply 2d-8 by d.

$$-12+6d=2d^{2}-8d$$

Subtract 2d^{2} from both sides.

$$-12+6d-2d^{2}=-8d$$

Add 8d to both sides.

$$-12+6d-2d^{2}+8d=0$$

Combine 6d and 8d to get 14d.

$$-12+14d-2d^{2}=0$$

Divide both sides by 2.

$$-6+7d-d^{2}=0$$

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

$$-d^{2}+7d-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -d^{2}+ad+bd-6. To find a and b, set up a system to be solved.

$$a+b=7 ab=-\left(-6\right)=6$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 6.

$$1,6 2,3$$

Calculate the sum for each pair.

$$1+6=7 2+3=5$$

The solution is the pair that gives sum 7.

$$a=6 b=1$$

Rewrite -d^{2}+7d-6 as \left(-d^{2}+6d\right)+\left(d-6\right).

$$\left(-d^{2}+6d\right)+\left(d-6\right)$$

Factor out -d in -d^{2}+6d.

$$-d\left(d-6\right)+d-6$$

Factor out common term d-6 by using distributive property.

$$\left(d-6\right)\left(-d+1\right)$$

To find equation solutions, solve d-6=0 and -d+1=0.

$$d=6 d=1$$

Steps Using the Quadratic Formula

Variable d cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by \left(d-4\right)\left(d-2\right), the least common multiple of 4-d,d-2.

$$\left(2-d\right)\left(-6\right)=\left(d-4\right)\times 2d$$

Use the distributive property to multiply 2-d by -6.

$$-12+6d=\left(d-4\right)\times 2d$$

Use the distributive property to multiply d-4 by 2.

$$-12+6d=\left(2d-8\right)d$$

Use the distributive property to multiply 2d-8 by d.

$$-12+6d=2d^{2}-8d$$

Subtract 2d^{2} from both sides.

$$-12+6d-2d^{2}=-8d$$

Add 8d to both sides.

$$-12+6d-2d^{2}+8d=0$$

Combine 6d and 8d to get 14d.

$$-12+14d-2d^{2}=0$$

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.

$$-2d^{2}+14d-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 14 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-14±\sqrt{14^{2}-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}$$

Square 14.

$$d=\frac{-14±\sqrt{196-4\left(-2\right)\left(-12\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$d=\frac{-14±\sqrt{196+8\left(-12\right)}}{2\left(-2\right)}$$

Multiply 8 times -12.

$$d=\frac{-14±\sqrt{196-96}}{2\left(-2\right)}$$

Add 196 to -96.

$$d=\frac{-14±\sqrt{100}}{2\left(-2\right)}$$

Take the square root of 100.

$$d=\frac{-14±10}{2\left(-2\right)}$$

Multiply 2 times -2.

$$d=\frac{-14±10}{-4}$$

Now solve the equation d=\frac{-14±10}{-4} when ± is plus. Add -14 to 10.

$$d=-\frac{4}{-4}$$

Divide -4 by -4.

$$d=1$$

Now solve the equation d=\frac{-14±10}{-4} when ± is minus. Subtract 10 from -14.

$$d=-\frac{24}{-4}$$

Divide -24 by -4.

$$d=6$$

The equation is now solved.

$$d=1 d=6$$

Steps for Completing the Square

Variable d cannot be equal to any of the values 2,4 since division by zero is not defined. Multiply both sides of the equation by \left(d-4\right)\left(d-2\right), the least common multiple of 4-d,d-2.

$$\left(2-d\right)\left(-6\right)=\left(d-4\right)\times 2d$$

Use the distributive property to multiply 2-d by -6.

$$-12+6d=\left(d-4\right)\times 2d$$

Use the distributive property to multiply d-4 by 2.

$$-12+6d=\left(2d-8\right)d$$

Use the distributive property to multiply 2d-8 by d.

$$-12+6d=2d^{2}-8d$$

Subtract 2d^{2} from both sides.

$$-12+6d-2d^{2}=-8d$$

Add 8d to both sides.

$$-12+6d-2d^{2}+8d=0$$

Combine 6d and 8d to get 14d.

$$-12+14d-2d^{2}=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$14d-2d^{2}=12$$

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.

$$-2d^{2}+14d=12$$

Divide both sides by -2.

$$\frac{-2d^{2}+14d}{-2}=\frac{12}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$d^{2}+\frac{14}{-2}d=\frac{12}{-2}$$

Divide 14 by -2.

$$d^{2}-7d=\frac{12}{-2}$$

Divide 12 by -2.

$$d^{2}-7d=-6$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-7d+\frac{49}{4}=-6+\frac{49}{4}$$

Add -6 to \frac{49}{4}.

$$d^{2}-7d+\frac{49}{4}=\frac{25}{4}$$

Factor d^{2}-7d+\frac{49}{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}.

$$\left(d-\frac{7}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{7}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$d-\frac{7}{2}=\frac{5}{2} d-\frac{7}{2}=-\frac{5}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$d=6 d=1$$