Solve 22/14-d=61+d/22

$$\frac { 22 } { 14 - d } = \frac { 61 + d } { 22 }$$

Solve for d

$$d = \frac{\sqrt{3689} - 47}{2} \approx 6.868569278$$

$$d=\frac{-\sqrt{3689}-47}{2}\approx -53.868569278$$

Steps Using the Quadratic Formula

Variable d cannot be equal to 14 since division by zero is not defined. Multiply both sides of the equation by 22\left(d-14\right), the least common multiple of 14-d,22.

$$-22\times 22=\left(d-14\right)\left(61+d\right)$$

Multiply -22 and 22 to get -484.

$$-484=\left(d-14\right)\left(61+d\right)$$

Use the distributive property to multiply d-14 by 61+d and combine like terms.

$$-484=47d+d^{2}-854$$

Swap sides so that all variable terms are on the left hand side.

$$47d+d^{2}-854=-484$$

Add 484 to both sides.

$$47d+d^{2}-854+484=0$$

Add -854 and 484 to get -370.

$$47d+d^{2}-370=0$$

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.

$$d^{2}+47d-370=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 47 for b, and -370 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-47±\sqrt{47^{2}-4\left(-370\right)}}{2}$$

Square 47.

$$d=\frac{-47±\sqrt{2209-4\left(-370\right)}}{2}$$

Multiply -4 times -370.

$$d=\frac{-47±\sqrt{2209+1480}}{2}$$

Add 2209 to 1480.

$$d=\frac{-47±\sqrt{3689}}{2}$$

Now solve the equation d=\frac{-47±\sqrt{3689}}{2} when ± is plus. Add -47 to \sqrt{3689}.

$$d=\frac{\sqrt{3689}-47}{2}$$

Now solve the equation d=\frac{-47±\sqrt{3689}}{2} when ± is minus. Subtract \sqrt{3689} from -47.

$$d=\frac{-\sqrt{3689}-47}{2}$$

The equation is now solved.

$$d=\frac{\sqrt{3689}-47}{2} d=\frac{-\sqrt{3689}-47}{2}$$

Steps for Completing the Square

Variable d cannot be equal to 14 since division by zero is not defined. Multiply both sides of the equation by 22\left(d-14\right), the least common multiple of 14-d,22.

$$-22\times 22=\left(d-14\right)\left(61+d\right)$$

Multiply -22 and 22 to get -484.

$$-484=\left(d-14\right)\left(61+d\right)$$

Use the distributive property to multiply d-14 by 61+d and combine like terms.

$$-484=47d+d^{2}-854$$

Swap sides so that all variable terms are on the left hand side.

$$47d+d^{2}-854=-484$$

Add 854 to both sides.

$$47d+d^{2}=-484+854$$

Add -484 and 854 to get 370.

$$47d+d^{2}=370$$

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.

$$d^{2}+47d=370$$

Divide 47, the coefficient of the x term, by 2 to get \frac{47}{2}. Then add the square of \frac{47}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+47d+\left(\frac{47}{2}\right)^{2}=370+\left(\frac{47}{2}\right)^{2}$$

Square \frac{47}{2} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+47d+\frac{2209}{4}=370+\frac{2209}{4}$$

Add 370 to \frac{2209}{4}.

$$d^{2}+47d+\frac{2209}{4}=\frac{3689}{4}$$

Factor d^{2}+47d+\frac{2209}{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}.

$$\left(d+\frac{47}{2}\right)^{2}=\frac{3689}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{47}{2}\right)^{2}}=\sqrt{\frac{3689}{4}}$$

Simplify.

$$d+\frac{47}{2}=\frac{\sqrt{3689}}{2} d+\frac{47}{2}=-\frac{\sqrt{3689}}{2}$$

Subtract \frac{47}{2} from both sides of the equation.

$$d=\frac{\sqrt{3689}-47}{2} d=\frac{-\sqrt{3689}-47}{2}$$