Solve m^2=8m+14

$$m ^ { 2 } = 8 m + 14$$

Solve for m

$$m=\sqrt{30}+4\approx 9.477225575$$

$$m=4-\sqrt{30}\approx -1.477225575$$

Steps Using the Quadratic Formula

Subtract 8m from both sides.

$$m^{2}-8m=14$$

Subtract 14 from both sides.

$$m^{2}-8m-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-14\right)}}{2}$$

Square -8.

$$m=\frac{-\left(-8\right)±\sqrt{64-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$m=\frac{-\left(-8\right)±\sqrt{64+56}}{2}$$

Add 64 to 56.

$$m=\frac{-\left(-8\right)±\sqrt{120}}{2}$$

Take the square root of 120.

$$m=\frac{-\left(-8\right)±2\sqrt{30}}{2}$$

The opposite of -8 is 8.

$$m=\frac{8±2\sqrt{30}}{2}$$

Now solve the equation m=\frac{8±2\sqrt{30}}{2} when ± is plus. Add 8 to 2\sqrt{30}.

$$m=\frac{2\sqrt{30}+8}{2}$$

Divide 8+2\sqrt{30} by 2.

$$m=\sqrt{30}+4$$

Now solve the equation m=\frac{8±2\sqrt{30}}{2} when ± is minus. Subtract 2\sqrt{30} from 8.

$$m=\frac{8-2\sqrt{30}}{2}$$

Divide 8-2\sqrt{30} by 2.

$$m=4-\sqrt{30}$$

The equation is now solved.

$$m=\sqrt{30}+4 m=4-\sqrt{30}$$

Steps for Completing the Square

Subtract 8m from both sides.

$$m^{2}-8m=14$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-8m+\left(-4\right)^{2}=14+\left(-4\right)^{2}$$

Square -4.

$$m^{2}-8m+16=14+16$$

Add 14 to 16.

$$m^{2}-8m+16=30$$

Factor m^{2}-8m+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-4\right)^{2}=30$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-4\right)^{2}}=\sqrt{30}$$

Simplify.

$$m-4=\sqrt{30} m-4=-\sqrt{30}$$

Add 4 to both sides of the equation.

$$m=\sqrt{30}+4 m=4-\sqrt{30}$$