Solve 140/d-140/d+7=1

$$\frac { 140 } { d } - \frac { 140 } { d + 7 } = 1$$

Solve for d

$$d=-35$$

$$d=28$$

Steps Using Factoring By Grouping

Variable d cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by d\left(d+7\right), the least common multiple of d,d+7.

$$\left(d+7\right)\times 140-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d+7 by 140.

$$140d+980-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d by d+7.

$$140d+980-d\times 140=d^{2}+7d$$

Subtract d^{2} from both sides.

$$140d+980-d\times 140-d^{2}=7d$$

Subtract 7d from both sides.

$$140d+980-d\times 140-d^{2}-7d=0$$

Combine 140d and -7d to get 133d.

$$133d+980-d\times 140-d^{2}=0$$

Multiply -1 and 140 to get -140.

$$133d+980-140d-d^{2}=0$$

Combine 133d and -140d to get -7d.

$$-7d+980-d^{2}=0$$

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

$$-d^{2}-7d+980=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -d^{2}+ad+bd+980. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-980=-980$$

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 -980.

$$1,-980 2,-490 4,-245 5,-196 7,-140 10,-98 14,-70 20,-49 28,-35$$

Calculate the sum for each pair.

$$1-980=-979 2-490=-488 4-245=-241 5-196=-191 7-140=-133 10-98=-88 14-70=-56 20-49=-29 28-35=-7$$

The solution is the pair that gives sum -7.

$$a=28 b=-35$$

Rewrite -d^{2}-7d+980 as \left(-d^{2}+28d\right)+\left(-35d+980\right).

$$\left(-d^{2}+28d\right)+\left(-35d+980\right)$$

Factor out d in the first and 35 in the second group.

$$d\left(-d+28\right)+35\left(-d+28\right)$$

Factor out common term -d+28 by using distributive property.

$$\left(-d+28\right)\left(d+35\right)$$

To find equation solutions, solve -d+28=0 and d+35=0.

$$d=28 d=-35$$

Steps Using the Quadratic Formula

Variable d cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by d\left(d+7\right), the least common multiple of d,d+7.

$$\left(d+7\right)\times 140-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d+7 by 140.

$$140d+980-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d by d+7.

$$140d+980-d\times 140=d^{2}+7d$$

Subtract d^{2} from both sides.

$$140d+980-d\times 140-d^{2}=7d$$

Subtract 7d from both sides.

$$140d+980-d\times 140-d^{2}-7d=0$$

Combine 140d and -7d to get 133d.

$$133d+980-d\times 140-d^{2}=0$$

Multiply -1 and 140 to get -140.

$$133d+980-140d-d^{2}=0$$

Combine 133d and -140d to get -7d.

$$-7d+980-d^{2}=0$$

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.

$$-d^{2}-7d+980=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -7 for b, and 980 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\times 980}}{2\left(-1\right)}$$

Square -7.

$$d=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\times 980}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$d=\frac{-\left(-7\right)±\sqrt{49+4\times 980}}{2\left(-1\right)}$$

Multiply 4 times 980.

$$d=\frac{-\left(-7\right)±\sqrt{49+3920}}{2\left(-1\right)}$$

Add 49 to 3920.

$$d=\frac{-\left(-7\right)±\sqrt{3969}}{2\left(-1\right)}$$

Take the square root of 3969.

$$d=\frac{-\left(-7\right)±63}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$d=\frac{7±63}{2\left(-1\right)}$$

Multiply 2 times -1.

$$d=\frac{7±63}{-2}$$

Now solve the equation d=\frac{7±63}{-2} when ± is plus. Add 7 to 63.

$$d=\frac{70}{-2}$$

Divide 70 by -2.

$$d=-35$$

Now solve the equation d=\frac{7±63}{-2} when ± is minus. Subtract 63 from 7.

$$d=-\frac{56}{-2}$$

Divide -56 by -2.

$$d=28$$

The equation is now solved.

$$d=-35 d=28$$

Steps for Completing the Square

Variable d cannot be equal to any of the values -7,0 since division by zero is not defined. Multiply both sides of the equation by d\left(d+7\right), the least common multiple of d,d+7.

$$\left(d+7\right)\times 140-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d+7 by 140.

$$140d+980-d\times 140=d\left(d+7\right)$$

Use the distributive property to multiply d by d+7.

$$140d+980-d\times 140=d^{2}+7d$$

Subtract d^{2} from both sides.

$$140d+980-d\times 140-d^{2}=7d$$

Subtract 7d from both sides.

$$140d+980-d\times 140-d^{2}-7d=0$$

Combine 140d and -7d to get 133d.

$$133d+980-d\times 140-d^{2}=0$$

Subtract 980 from both sides. Anything subtracted from zero gives its negation.

$$133d-d\times 140-d^{2}=-980$$

Multiply -1 and 140 to get -140.

$$133d-140d-d^{2}=-980$$

Combine 133d and -140d to get -7d.

$$-7d-d^{2}=-980$$

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.

$$-d^{2}-7d=-980$$

Divide both sides by -1.

$$\frac{-d^{2}-7d}{-1}=-\frac{980}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$d^{2}+\left(-\frac{7}{-1}\right)d=-\frac{980}{-1}$$

Divide -7 by -1.

$$d^{2}+7d=-\frac{980}{-1}$$

Divide -980 by -1.

$$d^{2}+7d=980$$

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.

$$d^{2}+7d+\left(\frac{7}{2}\right)^{2}=980+\left(\frac{7}{2}\right)^{2}$$

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

$$d^{2}+7d+\frac{49}{4}=980+\frac{49}{4}$$

Add 980 to \frac{49}{4}.

$$d^{2}+7d+\frac{49}{4}=\frac{3969}{4}$$

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}.

$$\left(d+\frac{7}{2}\right)^{2}=\frac{3969}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{7}{2}\right)^{2}}=\sqrt{\frac{3969}{4}}$$

Simplify.

$$d+\frac{7}{2}=\frac{63}{2} d+\frac{7}{2}=-\frac{63}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$d=28 d=-35$$