Solve (25+10d)^2=25(25+12d)

$$( 25 + 10 d ) ^ { 2 } = 25 ( 25 + 12 d )$$

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(25+10d\right)^{2}.

$$625+500d+100d^{2}=25\left(25+12d\right)$$

Use the distributive property to multiply 25 by 25+12d.

$$625+500d+100d^{2}=625+300d$$

Subtract 625 from both sides.

$$625+500d+100d^{2}-625=300d$$

Subtract 625 from 625 to get 0.

$$500d+100d^{2}=300d$$

Subtract 300d from both sides.

$$500d+100d^{2}-300d=0$$

Combine 500d and -300d to get 200d.

$$200d+100d^{2}=0$$

Factor out d.

$$d\left(200+100d\right)=0$$

To find equation solutions, solve d=0 and 200+100d=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(25+10d\right)^{2}.

$$625+500d+100d^{2}=25\left(25+12d\right)$$

Use the distributive property to multiply 25 by 25+12d.

$$625+500d+100d^{2}=625+300d$$

Subtract 625 from both sides.

$$625+500d+100d^{2}-625=300d$$

Subtract 625 from 625 to get 0.

$$500d+100d^{2}=300d$$

Subtract 300d from both sides.

$$500d+100d^{2}-300d=0$$

Combine 500d and -300d to get 200d.

$$200d+100d^{2}=0$$

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.

$$100d^{2}+200d=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 200 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-200±\sqrt{200^{2}}}{2\times 100}$$

Take the square root of 200^{2}.

$$d=\frac{-200±200}{2\times 100}$$

Multiply 2 times 100.

$$d=\frac{-200±200}{200}$$

Now solve the equation d=\frac{-200±200}{200} when ± is plus. Add -200 to 200.

$$d=\frac{0}{200}$$

Divide 0 by 200.

$$d=0$$

Now solve the equation d=\frac{-200±200}{200} when ± is minus. Subtract 200 from -200.

$$d=-\frac{400}{200}$$

Divide -400 by 200.

$$d=-2$$

The equation is now solved.

$$d=0 d=-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(25+10d\right)^{2}.

$$625+500d+100d^{2}=25\left(25+12d\right)$$

Use the distributive property to multiply 25 by 25+12d.

$$625+500d+100d^{2}=625+300d$$

Subtract 300d from both sides.

$$625+500d+100d^{2}-300d=625$$

Combine 500d and -300d to get 200d.

$$625+200d+100d^{2}=625$$

Subtract 625 from both sides.

$$200d+100d^{2}=625-625$$

Subtract 625 from 625 to get 0.

$$200d+100d^{2}=0$$

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.

$$100d^{2}+200d=0$$

Divide both sides by 100.

$$\frac{100d^{2}+200d}{100}=\frac{0}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$d^{2}+\frac{200}{100}d=\frac{0}{100}$$

Divide 200 by 100.

$$d^{2}+2d=\frac{0}{100}$$

Divide 0 by 100.

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

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.

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

Square 1.

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

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.

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

Subtract 1 from both sides of the equation.

$$d=0 d=-2$$