Solve (m+4)^2+m^2=6^2

$$( m + 4 ) ^ { 2 } + m ^ { 2 } = 6 ^ { 2 }$$

Solve for m

$$m=\sqrt{14}-2\approx 1.741657387$$

$$m=-\sqrt{14}-2\approx -5.741657387$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+4\right)^{2}.

$$m^{2}+8m+16+m^{2}=6^{2}$$

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

$$2m^{2}+8m+16=6^{2}$$

Calculate 6 to the power of 2 and get 36.

$$2m^{2}+8m+16=36$$

Subtract 36 from both sides.

$$2m^{2}+8m+16-36=0$$

Subtract 36 from 16 to get -20.

$$2m^{2}+8m-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-8±\sqrt{8^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Square 8.

$$m=\frac{-8±\sqrt{64-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$m=\frac{-8±\sqrt{64-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$m=\frac{-8±\sqrt{64+160}}{2\times 2}$$

Add 64 to 160.

$$m=\frac{-8±\sqrt{224}}{2\times 2}$$

Take the square root of 224.

$$m=\frac{-8±4\sqrt{14}}{2\times 2}$$

Multiply 2 times 2.

$$m=\frac{-8±4\sqrt{14}}{4}$$

Now solve the equation m=\frac{-8±4\sqrt{14}}{4} when ± is plus. Add -8 to 4\sqrt{14}.

$$m=\frac{4\sqrt{14}-8}{4}$$

Divide -8+4\sqrt{14} by 4.

$$m=\sqrt{14}-2$$

Now solve the equation m=\frac{-8±4\sqrt{14}}{4} when ± is minus. Subtract 4\sqrt{14} from -8.

$$m=\frac{-4\sqrt{14}-8}{4}$$

Divide -8-4\sqrt{14} by 4.

$$m=-\sqrt{14}-2$$

The equation is now solved.

$$m=\sqrt{14}-2 m=-\sqrt{14}-2$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(m+4\right)^{2}.

$$m^{2}+8m+16+m^{2}=6^{2}$$

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

$$2m^{2}+8m+16=6^{2}$$

Calculate 6 to the power of 2 and get 36.

$$2m^{2}+8m+16=36$$

Subtract 16 from both sides.

$$2m^{2}+8m=36-16$$

Subtract 16 from 36 to get 20.

$$2m^{2}+8m=20$$

Divide both sides by 2.

$$\frac{2m^{2}+8m}{2}=\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$m^{2}+\frac{8}{2}m=\frac{20}{2}$$

Divide 8 by 2.

$$m^{2}+4m=\frac{20}{2}$$

Divide 20 by 2.

$$m^{2}+4m=10$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}+4m+2^{2}=10+2^{2}$$

Square 2.

$$m^{2}+4m+4=10+4$$

Add 10 to 4.

$$m^{2}+4m+4=14$$

Factor m^{2}+4m+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m+2\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m+2\right)^{2}}=\sqrt{14}$$

Simplify.

$$m+2=\sqrt{14} m+2=-\sqrt{14}$$

Subtract 2 from both sides of the equation.

$$m=\sqrt{14}-2 m=-\sqrt{14}-2$$