Solve 0=(-8-6m)^2-4(1+m^2)(16)

$$0 = ( - 8 - 6 m ) ^ { 2 } - 4 ( 1 + m ^ { 2 } ) ( 16 )$$

Solve for m

$$m = \frac{24}{7} = 3\frac{3}{7} \approx 3.428571429$$

$$m=0$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-8-6m\right)^{2}.

$$0=64+96m+36m^{2}-4\left(1+m^{2}\right)\times 16$$

Multiply 4 and 16 to get 64.

$$0=64+96m+36m^{2}-64\left(1+m^{2}\right)$$

Use the distributive property to multiply -64 by 1+m^{2}.

$$0=64+96m+36m^{2}-64-64m^{2}$$

Subtract 64 from 64 to get 0.

$$0=96m+36m^{2}-64m^{2}$$

Combine 36m^{2} and -64m^{2} to get -28m^{2}.

$$0=96m-28m^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$96m-28m^{2}=0$$

Factor out m.

$$m\left(96-28m\right)=0$$

To find equation solutions, solve m=0 and 96-28m=0.

$$m=0 m=\frac{24}{7}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-8-6m\right)^{2}.

$$0=64+96m+36m^{2}-4\left(1+m^{2}\right)\times 16$$

Multiply 4 and 16 to get 64.

$$0=64+96m+36m^{2}-64\left(1+m^{2}\right)$$

Use the distributive property to multiply -64 by 1+m^{2}.

$$0=64+96m+36m^{2}-64-64m^{2}$$

Subtract 64 from 64 to get 0.

$$0=96m+36m^{2}-64m^{2}$$

Combine 36m^{2} and -64m^{2} to get -28m^{2}.

$$0=96m-28m^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$96m-28m^{2}=0$$

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.

$$-28m^{2}+96m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -28 for a, 96 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-96±\sqrt{96^{2}}}{2\left(-28\right)}$$

Take the square root of 96^{2}.

$$m=\frac{-96±96}{2\left(-28\right)}$$

Multiply 2 times -28.

$$m=\frac{-96±96}{-56}$$

Now solve the equation m=\frac{-96±96}{-56} when ± is plus. Add -96 to 96.

$$m=\frac{0}{-56}$$

Divide 0 by -56.

$$m=0$$

Now solve the equation m=\frac{-96±96}{-56} when ± is minus. Subtract 96 from -96.

$$m=-\frac{192}{-56}$$

Reduce the fraction \frac{-192}{-56} to lowest terms by extracting and canceling out 8.

$$m=\frac{24}{7}$$

The equation is now solved.

$$m=0 m=\frac{24}{7}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-8-6m\right)^{2}.

$$0=64+96m+36m^{2}-4\left(1+m^{2}\right)\times 16$$

Multiply 4 and 16 to get 64.

$$0=64+96m+36m^{2}-64\left(1+m^{2}\right)$$

Use the distributive property to multiply -64 by 1+m^{2}.

$$0=64+96m+36m^{2}-64-64m^{2}$$

Subtract 64 from 64 to get 0.

$$0=96m+36m^{2}-64m^{2}$$

Combine 36m^{2} and -64m^{2} to get -28m^{2}.

$$0=96m-28m^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$96m-28m^{2}=0$$

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.

$$-28m^{2}+96m=0$$

Divide both sides by -28.

$$\frac{-28m^{2}+96m}{-28}=\frac{0}{-28}$$

Dividing by -28 undoes the multiplication by -28.

$$m^{2}+\frac{96}{-28}m=\frac{0}{-28}$$

Reduce the fraction \frac{96}{-28} to lowest terms by extracting and canceling out 4.

$$m^{2}-\frac{24}{7}m=\frac{0}{-28}$$

Divide 0 by -28.

$$m^{2}-\frac{24}{7}m=0$$

Divide -\frac{24}{7}, the coefficient of the x term, by 2 to get -\frac{12}{7}. Then add the square of -\frac{12}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{24}{7}m+\left(-\frac{12}{7}\right)^{2}=\left(-\frac{12}{7}\right)^{2}$$

Square -\frac{12}{7} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{24}{7}m+\frac{144}{49}=\frac{144}{49}$$

Factor m^{2}-\frac{24}{7}m+\frac{144}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{12}{7}\right)^{2}=\frac{144}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{12}{7}\right)^{2}}=\sqrt{\frac{144}{49}}$$

Simplify.

$$m-\frac{12}{7}=\frac{12}{7} m-\frac{12}{7}=-\frac{12}{7}$$

Add \frac{12}{7} to both sides of the equation.

$$m=\frac{24}{7} m=0$$