Solve 100=(m-8)^2+(3/4m-6)^2

$$100 = ( m - 8 ) ^ { 2 } + ( \frac { 3 } { 4 } m - 6 ) ^ { 2 }$$

Solve for m

$$m=16$$

$$m=0$$

Steps Using Factoring

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

$$100=m^{2}-16m+64+\left(\frac{3}{4}m-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}m-6\right)^{2}.

$$100=m^{2}-16m+64+\frac{9}{16}m^{2}-9m+36$$

Combine m^{2} and \frac{9}{16}m^{2} to get \frac{25}{16}m^{2}.

$$100=\frac{25}{16}m^{2}-16m+64-9m+36$$

Combine -16m and -9m to get -25m.

$$100=\frac{25}{16}m^{2}-25m+64+36$$

Add 64 and 36 to get 100.

$$100=\frac{25}{16}m^{2}-25m+100$$

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

$$\frac{25}{16}m^{2}-25m+100=100$$

Subtract 100 from both sides.

$$\frac{25}{16}m^{2}-25m+100-100=0$$

Subtract 100 from 100 to get 0.

$$\frac{25}{16}m^{2}-25m=0$$

Factor out m.

$$m\left(\frac{25}{16}m-25\right)=0$$

To find equation solutions, solve m=0 and \frac{25m}{16}-25=0.

$$m=0 m=16$$

Steps Using the Quadratic Formula

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

$$100=m^{2}-16m+64+\left(\frac{3}{4}m-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}m-6\right)^{2}.

$$100=m^{2}-16m+64+\frac{9}{16}m^{2}-9m+36$$

Combine m^{2} and \frac{9}{16}m^{2} to get \frac{25}{16}m^{2}.

$$100=\frac{25}{16}m^{2}-16m+64-9m+36$$

Combine -16m and -9m to get -25m.

$$100=\frac{25}{16}m^{2}-25m+64+36$$

Add 64 and 36 to get 100.

$$100=\frac{25}{16}m^{2}-25m+100$$

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

$$\frac{25}{16}m^{2}-25m+100=100$$

Subtract 100 from both sides.

$$\frac{25}{16}m^{2}-25m+100-100=0$$

Subtract 100 from 100 to get 0.

$$\frac{25}{16}m^{2}-25m=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{25}{16} for a, -25 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}}}{2\times \frac{25}{16}}$$

Take the square root of \left(-25\right)^{2}.

$$m=\frac{-\left(-25\right)±25}{2\times \frac{25}{16}}$$

The opposite of -25 is 25.

$$m=\frac{25±25}{2\times \frac{25}{16}}$$

Multiply 2 times \frac{25}{16}.

$$m=\frac{25±25}{\frac{25}{8}}$$

Now solve the equation m=\frac{25±25}{\frac{25}{8}} when ± is plus. Add 25 to 25.

$$m=\frac{50}{\frac{25}{8}}$$

Divide 50 by \frac{25}{8} by multiplying 50 by the reciprocal of \frac{25}{8}.

$$m=16$$

Now solve the equation m=\frac{25±25}{\frac{25}{8}} when ± is minus. Subtract 25 from 25.

$$m=\frac{0}{\frac{25}{8}}$$

Divide 0 by \frac{25}{8} by multiplying 0 by the reciprocal of \frac{25}{8}.

$$m=0$$

The equation is now solved.

$$m=16 m=0$$

Steps for Completing the Square

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

$$100=m^{2}-16m+64+\left(\frac{3}{4}m-6\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{3}{4}m-6\right)^{2}.

$$100=m^{2}-16m+64+\frac{9}{16}m^{2}-9m+36$$

Combine m^{2} and \frac{9}{16}m^{2} to get \frac{25}{16}m^{2}.

$$100=\frac{25}{16}m^{2}-16m+64-9m+36$$

Combine -16m and -9m to get -25m.

$$100=\frac{25}{16}m^{2}-25m+64+36$$

Add 64 and 36 to get 100.

$$100=\frac{25}{16}m^{2}-25m+100$$

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

$$\frac{25}{16}m^{2}-25m+100=100$$

Subtract 100 from both sides.

$$\frac{25}{16}m^{2}-25m=100-100$$

Subtract 100 from 100 to get 0.

$$\frac{25}{16}m^{2}-25m=0$$

Divide both sides of the equation by \frac{25}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{25}{16}m^{2}-25m}{\frac{25}{16}}=\frac{0}{\frac{25}{16}}$$

Dividing by \frac{25}{16} undoes the multiplication by \frac{25}{16}.

$$m^{2}+\left(-\frac{25}{\frac{25}{16}}\right)m=\frac{0}{\frac{25}{16}}$$

Divide -25 by \frac{25}{16} by multiplying -25 by the reciprocal of \frac{25}{16}.

$$m^{2}-16m=\frac{0}{\frac{25}{16}}$$

Divide 0 by \frac{25}{16} by multiplying 0 by the reciprocal of \frac{25}{16}.

$$m^{2}-16m=0$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-16m+\left(-8\right)^{2}=\left(-8\right)^{2}$$

Square -8.

$$m^{2}-16m+64=64$$

Factor m^{2}-16m+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-8\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-8\right)^{2}}=\sqrt{64}$$

Simplify.

$$m-8=8 m-8=-8$$

Add 8 to both sides of the equation.

$$m=16 m=0$$