Solve (3+r)^2+(15+r)^2=18^2

$$( 3 + r ) ^ { 2 } + ( 15 + r ) ^ { 2 } = 18 ^ { 2 }$$

Solve for r

$$r=3\sqrt{14}-9\approx 2.22497216$$

$$r=-3\sqrt{14}-9\approx -20.22497216$$

Steps Using the Quadratic Formula

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

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(15+r\right)^{2}.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

Add 9 and 225 to get 234.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

Combine 6r and 30r to get 36r.

$$234+36r+r^{2}+r^{2}=18^{2}$$

Combine r^{2} and r^{2} to get 2r^{2}.

$$234+36r+2r^{2}=18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$234+36r+2r^{2}=324$$

Subtract 324 from both sides.

$$234+36r+2r^{2}-324=0$$

Subtract 324 from 234 to get -90.

$$-90+36r+2r^{2}=0$$

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.

$$2r^{2}+36r-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 36 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-36±\sqrt{36^{2}-4\times 2\left(-90\right)}}{2\times 2}$$

Square 36.

$$r=\frac{-36±\sqrt{1296-4\times 2\left(-90\right)}}{2\times 2}$$

Multiply -4 times 2.

$$r=\frac{-36±\sqrt{1296-8\left(-90\right)}}{2\times 2}$$

Multiply -8 times -90.

$$r=\frac{-36±\sqrt{1296+720}}{2\times 2}$$

Add 1296 to 720.

$$r=\frac{-36±\sqrt{2016}}{2\times 2}$$

Take the square root of 2016.

$$r=\frac{-36±12\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$r=\frac{-36±12\sqrt{14}}{4}$$

Now solve the equation r=\frac{-36±12\sqrt{14}}{4} when ± is plus. Add -36 to 12\sqrt{14}.

$$r=\frac{12\sqrt{14}-36}{4}$$

Divide -36+12\sqrt{14} by 4.

$$r=3\sqrt{14}-9$$

Now solve the equation r=\frac{-36±12\sqrt{14}}{4} when ± is minus. Subtract 12\sqrt{14} from -36.

$$r=\frac{-12\sqrt{14}-36}{4}$$

Divide -36-12\sqrt{14} by 4.

$$r=-3\sqrt{14}-9$$

The equation is now solved.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$

Steps for Completing the Square

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

$$9+6r+r^{2}+\left(15+r\right)^{2}=18^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(15+r\right)^{2}.

$$9+6r+r^{2}+225+30r+r^{2}=18^{2}$$

Add 9 and 225 to get 234.

$$234+6r+r^{2}+30r+r^{2}=18^{2}$$

Combine 6r and 30r to get 36r.

$$234+36r+r^{2}+r^{2}=18^{2}$$

Combine r^{2} and r^{2} to get 2r^{2}.

$$234+36r+2r^{2}=18^{2}$$

Calculate 18 to the power of 2 and get 324.

$$234+36r+2r^{2}=324$$

Subtract 234 from both sides.

$$36r+2r^{2}=324-234$$

Subtract 234 from 324 to get 90.

$$36r+2r^{2}=90$$

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.

$$2r^{2}+36r=90$$

Divide both sides by 2.

$$\frac{2r^{2}+36r}{2}=\frac{90}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$r^{2}+\frac{36}{2}r=\frac{90}{2}$$

Divide 36 by 2.

$$r^{2}+18r=\frac{90}{2}$$

Divide 90 by 2.

$$r^{2}+18r=45$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+18r+9^{2}=45+9^{2}$$

Square 9.

$$r^{2}+18r+81=45+81$$

Add 45 to 81.

$$r^{2}+18r+81=126$$

Factor r^{2}+18r+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+9\right)^{2}=126$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+9\right)^{2}}=\sqrt{126}$$

Simplify.

$$r+9=3\sqrt{14} r+9=-3\sqrt{14}$$

Subtract 9 from both sides of the equation.

$$r=3\sqrt{14}-9 r=-3\sqrt{14}-9$$