Solve (10+2d)^2=(10+d)/frac{1{10+6d}}

$$( 10 + 2 d ) ^ { 2 } = \frac { ( 10 + d ) } { \frac { 1 } { 10 + 6 d } }$$

Solve for d

$$d=-15$$

$$d=0$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+2d\right)^{2}.

$$100+40d+4d^{2}=\frac{10+d}{\frac{1}{10+6d}}$$

Variable d cannot be equal to -\frac{5}{3} since division by zero is not defined. Divide 10+d by \frac{1}{10+6d} by multiplying 10+d by the reciprocal of \frac{1}{10+6d}.

$$100+40d+4d^{2}=\left(10+d\right)\left(10+6d\right)$$

Use the distributive property to multiply 10+d by 10+6d and combine like terms.

$$100+40d+4d^{2}=100+70d+6d^{2}$$

Subtract 100 from both sides.

$$100+40d+4d^{2}-100=70d+6d^{2}$$

Subtract 100 from 100 to get 0.

$$40d+4d^{2}=70d+6d^{2}$$

Subtract 70d from both sides.

$$40d+4d^{2}-70d=6d^{2}$$

Combine 40d and -70d to get -30d.

$$-30d+4d^{2}=6d^{2}$$

Subtract 6d^{2} from both sides.

$$-30d+4d^{2}-6d^{2}=0$$

Combine 4d^{2} and -6d^{2} to get -2d^{2}.

$$-30d-2d^{2}=0$$

Factor out d.

$$d\left(-30-2d\right)=0$$

To find equation solutions, solve d=0 and -30-2d=0.

$$d=0 d=-15$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+2d\right)^{2}.

$$100+40d+4d^{2}=\frac{10+d}{\frac{1}{10+6d}}$$

Variable d cannot be equal to -\frac{5}{3} since division by zero is not defined. Divide 10+d by \frac{1}{10+6d} by multiplying 10+d by the reciprocal of \frac{1}{10+6d}.

$$100+40d+4d^{2}=\left(10+d\right)\left(10+6d\right)$$

Use the distributive property to multiply 10+d by 10+6d and combine like terms.

$$100+40d+4d^{2}=100+70d+6d^{2}$$

Subtract 100 from both sides.

$$100+40d+4d^{2}-100=70d+6d^{2}$$

Subtract 100 from 100 to get 0.

$$40d+4d^{2}=70d+6d^{2}$$

Subtract 70d from both sides.

$$40d+4d^{2}-70d=6d^{2}$$

Combine 40d and -70d to get -30d.

$$-30d+4d^{2}=6d^{2}$$

Subtract 6d^{2} from both sides.

$$-30d+4d^{2}-6d^{2}=0$$

Combine 4d^{2} and -6d^{2} to get -2d^{2}.

$$-30d-2d^{2}=0$$

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.

$$-2d^{2}-30d=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -30 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}}}{2\left(-2\right)}$$

Take the square root of \left(-30\right)^{2}.

$$d=\frac{-\left(-30\right)±30}{2\left(-2\right)}$$

The opposite of -30 is 30.

$$d=\frac{30±30}{2\left(-2\right)}$$

Multiply 2 times -2.

$$d=\frac{30±30}{-4}$$

Now solve the equation d=\frac{30±30}{-4} when ± is plus. Add 30 to 30.

$$d=\frac{60}{-4}$$

Divide 60 by -4.

$$d=-15$$

Now solve the equation d=\frac{30±30}{-4} when ± is minus. Subtract 30 from 30.

$$d=\frac{0}{-4}$$

Divide 0 by -4.

$$d=0$$

The equation is now solved.

$$d=-15 d=0$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+2d\right)^{2}.

$$100+40d+4d^{2}=\frac{10+d}{\frac{1}{10+6d}}$$

Variable d cannot be equal to -\frac{5}{3} since division by zero is not defined. Divide 10+d by \frac{1}{10+6d} by multiplying 10+d by the reciprocal of \frac{1}{10+6d}.

$$100+40d+4d^{2}=\left(10+d\right)\left(10+6d\right)$$

Use the distributive property to multiply 10+d by 10+6d and combine like terms.

$$100+40d+4d^{2}=100+70d+6d^{2}$$

Subtract 70d from both sides.

$$100+40d+4d^{2}-70d=100+6d^{2}$$

Combine 40d and -70d to get -30d.

$$100-30d+4d^{2}=100+6d^{2}$$

Subtract 6d^{2} from both sides.

$$100-30d+4d^{2}-6d^{2}=100$$

Combine 4d^{2} and -6d^{2} to get -2d^{2}.

$$100-30d-2d^{2}=100$$

Subtract 100 from both sides.

$$-30d-2d^{2}=100-100$$

Subtract 100 from 100 to get 0.

$$-30d-2d^{2}=0$$

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.

$$-2d^{2}-30d=0$$

Divide both sides by -2.

$$\frac{-2d^{2}-30d}{-2}=\frac{0}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$d^{2}+\left(-\frac{30}{-2}\right)d=\frac{0}{-2}$$

Divide -30 by -2.

$$d^{2}+15d=\frac{0}{-2}$$

Divide 0 by -2.

$$d^{2}+15d=0$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}+15d+\left(\frac{15}{2}\right)^{2}=\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$d^{2}+15d+\frac{225}{4}=\frac{225}{4}$$

Factor d^{2}+15d+\frac{225}{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}.

$$\left(d+\frac{15}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d+\frac{15}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$d+\frac{15}{2}=\frac{15}{2} d+\frac{15}{2}=-\frac{15}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$d=0 d=-15$$