Solve 1=(2m-4)^2-2(m-4)

$$1 = ( 2 m - 4 ) ^ { 2 } - 2 ( m - 4 )$$

Solve for m

$$m=\frac{-\sqrt{11}i+9}{4}\approx 2.25-0.829156198i$$

$$m=\frac{9+\sqrt{11}i}{4}\approx 2.25+0.829156198i$$

Steps Using the Quadratic Formula

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

$$1=4m^{2}-16m+16-2\left(m-4\right)$$

Use the distributive property to multiply -2 by m-4.

$$1=4m^{2}-16m+16-2m+8$$

Combine -16m and -2m to get -18m.

$$1=4m^{2}-18m+16+8$$

Add 16 and 8 to get 24.

$$1=4m^{2}-18m+24$$

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

$$4m^{2}-18m+24=1$$

Subtract 1 from both sides.

$$4m^{2}-18m+24-1=0$$

Subtract 1 from 24 to get 23.

$$4m^{2}-18m+23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -18 for b, and 23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 4\times 23}}{2\times 4}$$

Square -18.

$$m=\frac{-\left(-18\right)±\sqrt{324-4\times 4\times 23}}{2\times 4}$$

Multiply -4 times 4.

$$m=\frac{-\left(-18\right)±\sqrt{324-16\times 23}}{2\times 4}$$

Multiply -16 times 23.

$$m=\frac{-\left(-18\right)±\sqrt{324-368}}{2\times 4}$$

Add 324 to -368.

$$m=\frac{-\left(-18\right)±\sqrt{-44}}{2\times 4}$$

Take the square root of -44.

$$m=\frac{-\left(-18\right)±2\sqrt{11}i}{2\times 4}$$

The opposite of -18 is 18.

$$m=\frac{18±2\sqrt{11}i}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{18±2\sqrt{11}i}{8}$$

Now solve the equation m=\frac{18±2\sqrt{11}i}{8} when ± is plus. Add 18 to 2i\sqrt{11}.

$$m=\frac{18+2\sqrt{11}i}{8}$$

Divide 18+2i\sqrt{11} by 8.

$$m=\frac{9+\sqrt{11}i}{4}$$

Now solve the equation m=\frac{18±2\sqrt{11}i}{8} when ± is minus. Subtract 2i\sqrt{11} from 18.

$$m=\frac{-2\sqrt{11}i+18}{8}$$

Divide 18-2i\sqrt{11} by 8.

$$m=\frac{-\sqrt{11}i+9}{4}$$

The equation is now solved.

$$m=\frac{9+\sqrt{11}i}{4} m=\frac{-\sqrt{11}i+9}{4}$$

Steps for Completing the Square

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

$$1=4m^{2}-16m+16-2\left(m-4\right)$$

Use the distributive property to multiply -2 by m-4.

$$1=4m^{2}-16m+16-2m+8$$

Combine -16m and -2m to get -18m.

$$1=4m^{2}-18m+16+8$$

Add 16 and 8 to get 24.

$$1=4m^{2}-18m+24$$

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

$$4m^{2}-18m+24=1$$

Subtract 24 from both sides.

$$4m^{2}-18m=1-24$$

Subtract 24 from 1 to get -23.

$$4m^{2}-18m=-23$$

Divide both sides by 4.

$$\frac{4m^{2}-18m}{4}=-\frac{23}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}+\left(-\frac{18}{4}\right)m=-\frac{23}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

$$m^{2}-\frac{9}{2}m=-\frac{23}{4}$$

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{9}{2}m+\left(-\frac{9}{4}\right)^{2}=-\frac{23}{4}+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{9}{2}m+\frac{81}{16}=-\frac{23}{4}+\frac{81}{16}$$

Add -\frac{23}{4} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{9}{2}m+\frac{81}{16}=-\frac{11}{16}$$

Factor m^{2}-\frac{9}{2}m+\frac{81}{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}.

$$\left(m-\frac{9}{4}\right)^{2}=-\frac{11}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{9}{4}\right)^{2}}=\sqrt{-\frac{11}{16}}$$

Simplify.

$$m-\frac{9}{4}=\frac{\sqrt{11}i}{4} m-\frac{9}{4}=-\frac{\sqrt{11}i}{4}$$

Add \frac{9}{4} to both sides of the equation.

$$m=\frac{9+\sqrt{11}i}{4} m=\frac{-\sqrt{11}i+9}{4}$$