Solve sqrt{225-b^2}=21-b

$$\sqrt { 225 - b ^ { 2 } } = 21 - b$$

Solve for b

$$b=12$$

$$b=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{225-b^{2}}\right)^{2}=\left(21-b\right)^{2}$$

Calculate \sqrt{225-b^{2}} to the power of 2 and get 225-b^{2}.

$$225-b^{2}=\left(21-b\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(21-b\right)^{2}.

$$225-b^{2}=441-42b+b^{2}$$

Subtract 441 from both sides.

$$225-b^{2}-441=-42b+b^{2}$$

Subtract 441 from 225 to get -216.

$$-216-b^{2}=-42b+b^{2}$$

Add 42b to both sides.

$$-216-b^{2}+42b=b^{2}$$

Subtract b^{2} from both sides.

$$-216-b^{2}+42b-b^{2}=0$$

Combine -b^{2} and -b^{2} to get -2b^{2}.

$$-216-2b^{2}+42b=0$$

Divide both sides by 2.

$$-108-b^{2}+21b=0$$

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

$$-b^{2}+21b-108=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -b^{2}+ab+bb-108. To find a and b, set up a system to be solved.

$$a+b=21 ab=-\left(-108\right)=108$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 108.

$$1,108 2,54 3,36 4,27 6,18 9,12$$

Calculate the sum for each pair.

$$1+108=109 2+54=56 3+36=39 4+27=31 6+18=24 9+12=21$$

The solution is the pair that gives sum 21.

$$a=12 b=9$$

Rewrite -b^{2}+21b-108 as \left(-b^{2}+12b\right)+\left(9b-108\right).

$$\left(-b^{2}+12b\right)+\left(9b-108\right)$$

Factor out -b in the first and 9 in the second group.

$$-b\left(b-12\right)+9\left(b-12\right)$$

Factor out common term b-12 by using distributive property.

$$\left(b-12\right)\left(-b+9\right)$$

To find equation solutions, solve b-12=0 and -b+9=0.

$$b=12 b=9$$

Substitute 12 for b in the equation \sqrt{225-b^{2}}=21-b.

$$\sqrt{225-12^{2}}=21-12$$

Simplify. The value b=12 satisfies the equation.

$$9=9$$

Substitute 9 for b in the equation \sqrt{225-b^{2}}=21-b.

$$\sqrt{225-9^{2}}=21-9$$

Simplify. The value b=9 satisfies the equation.

$$12=12$$

List all solutions of \sqrt{225-b^{2}}=21-b.

$$b=12 b=9$$