Solve (L+3)^2+3^2=15^2

$$( L + 3 ) ^ { 2 } + 3 ^ { 2 } = 15 ^ { 2 }$$

Solve for L

$$L=6\sqrt{6}-3\approx 11.696938457$$

$$L=-6\sqrt{6}-3\approx -17.696938457$$

Steps Using the Quadratic Formula

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

$$L^{2}+6L+9+3^{2}=15^{2}$$

Calculate 3 to the power of 2 and get 9.

$$L^{2}+6L+9+9=15^{2}$$

Add 9 and 9 to get 18.

$$L^{2}+6L+18=15^{2}$$

Calculate 15 to the power of 2 and get 225.

$$L^{2}+6L+18=225$$

Subtract 225 from both sides.

$$L^{2}+6L+18-225=0$$

Subtract 225 from 18 to get -207.

$$L^{2}+6L-207=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -207 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$L=\frac{-6±\sqrt{6^{2}-4\left(-207\right)}}{2}$$

Square 6.

$$L=\frac{-6±\sqrt{36-4\left(-207\right)}}{2}$$

Multiply -4 times -207.

$$L=\frac{-6±\sqrt{36+828}}{2}$$

Add 36 to 828.

$$L=\frac{-6±\sqrt{864}}{2}$$

Take the square root of 864.

$$L=\frac{-6±12\sqrt{6}}{2}$$

Now solve the equation L=\frac{-6±12\sqrt{6}}{2} when ± is plus. Add -6 to 12\sqrt{6}.

$$L=\frac{12\sqrt{6}-6}{2}$$

Divide -6+12\sqrt{6} by 2.

$$L=6\sqrt{6}-3$$

Now solve the equation L=\frac{-6±12\sqrt{6}}{2} when ± is minus. Subtract 12\sqrt{6} from -6.

$$L=\frac{-12\sqrt{6}-6}{2}$$

Divide -6-12\sqrt{6} by 2.

$$L=-6\sqrt{6}-3$$

The equation is now solved.

$$L=6\sqrt{6}-3 L=-6\sqrt{6}-3$$

Steps for Completing the Square

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

$$L^{2}+6L+9+3^{2}=15^{2}$$

Calculate 3 to the power of 2 and get 9.

$$L^{2}+6L+9+9=15^{2}$$

Add 9 and 9 to get 18.

$$L^{2}+6L+18=15^{2}$$

Calculate 15 to the power of 2 and get 225.

$$L^{2}+6L+18=225$$

Subtract 18 from both sides.

$$L^{2}+6L=225-18$$

Subtract 18 from 225 to get 207.

$$L^{2}+6L=207$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$L^{2}+6L+3^{2}=207+3^{2}$$

Square 3.

$$L^{2}+6L+9=207+9$$

Add 207 to 9.

$$L^{2}+6L+9=216$$

Factor L^{2}+6L+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(L+3\right)^{2}=216$$

Take the square root of both sides of the equation.

$$\sqrt{\left(L+3\right)^{2}}=\sqrt{216}$$

Simplify.

$$L+3=6\sqrt{6} L+3=-6\sqrt{6}$$

Subtract 3 from both sides of the equation.

$$L=6\sqrt{6}-3 L=-6\sqrt{6}-3$$