Solve (2+3d)^2=(2+d)(2+7d)

$$( 2 + 3 d ) ^ { 2 } = ( 2 + d ) ( 2 + 7 d )$$

Solve for d

$$d=2$$

$$d=0$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+3d\right)^{2}.

$$4+12d+9d^{2}=\left(2+d\right)\left(2+7d\right)$$

Use the distributive property to multiply 2+d by 2+7d and combine like terms.

$$4+12d+9d^{2}=4+16d+7d^{2}$$

Subtract 4 from both sides.

$$4+12d+9d^{2}-4=16d+7d^{2}$$

Subtract 4 from 4 to get 0.

$$12d+9d^{2}=16d+7d^{2}$$

Subtract 16d from both sides.

$$12d+9d^{2}-16d=7d^{2}$$

Combine 12d and -16d to get -4d.

$$-4d+9d^{2}=7d^{2}$$

Subtract 7d^{2} from both sides.

$$-4d+9d^{2}-7d^{2}=0$$

Combine 9d^{2} and -7d^{2} to get 2d^{2}.

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

Factor out d.

$$d\left(-4+2d\right)=0$$

To find equation solutions, solve d=0 and -4+2d=0.

$$d=0 d=2$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+3d\right)^{2}.

$$4+12d+9d^{2}=\left(2+d\right)\left(2+7d\right)$$

Use the distributive property to multiply 2+d by 2+7d and combine like terms.

$$4+12d+9d^{2}=4+16d+7d^{2}$$

Subtract 4 from both sides.

$$4+12d+9d^{2}-4=16d+7d^{2}$$

Subtract 4 from 4 to get 0.

$$12d+9d^{2}=16d+7d^{2}$$

Subtract 16d from both sides.

$$12d+9d^{2}-16d=7d^{2}$$

Combine 12d and -16d to get -4d.

$$-4d+9d^{2}=7d^{2}$$

Subtract 7d^{2} from both sides.

$$-4d+9d^{2}-7d^{2}=0$$

Combine 9d^{2} and -7d^{2} to get 2d^{2}.

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

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.

$$2d^{2}-4d=0$$

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

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

Take the square root of \left(-4\right)^{2}.

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

The opposite of -4 is 4.

$$d=\frac{4±4}{2\times 2}$$

Multiply 2 times 2.

$$d=\frac{4±4}{4}$$

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

$$d=\frac{8}{4}$$

Divide 8 by 4.

$$d=2$$

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

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

Divide 0 by 4.

$$d=0$$

The equation is now solved.

$$d=2 d=0$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+3d\right)^{2}.

$$4+12d+9d^{2}=\left(2+d\right)\left(2+7d\right)$$

Use the distributive property to multiply 2+d by 2+7d and combine like terms.

$$4+12d+9d^{2}=4+16d+7d^{2}$$

Subtract 16d from both sides.

$$4+12d+9d^{2}-16d=4+7d^{2}$$

Combine 12d and -16d to get -4d.

$$4-4d+9d^{2}=4+7d^{2}$$

Subtract 7d^{2} from both sides.

$$4-4d+9d^{2}-7d^{2}=4$$

Combine 9d^{2} and -7d^{2} to get 2d^{2}.

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

Subtract 4 from both sides.

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

Subtract 4 from 4 to get 0.

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

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.

$$2d^{2}-4d=0$$

Divide both sides by 2.

$$\frac{2d^{2}-4d}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$d^{2}+\left(-\frac{4}{2}\right)d=\frac{0}{2}$$

Divide -4 by 2.

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

Divide 0 by 2.

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

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

$$d^{2}-2d+1=1$$

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

$$\left(d-1\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-1\right)^{2}}=\sqrt{1}$$

Simplify.

$$d-1=1 d-1=-1$$

Add 1 to both sides of the equation.

$$d=2 d=0$$