Solve (2m-3)(-(m-1))-3+1=0

$$( 2 m - 3 ) ( - ( m - 1 ) ) - 3 + 1 = 0$$

Solve for m

$$m=\frac{-\sqrt{15}i+5}{4}\approx 1.25-0.968245837i$$

$$m=\frac{5+\sqrt{15}i}{4}\approx 1.25+0.968245837i$$

Steps Using the Quadratic Formula

To find the opposite of m-1, find the opposite of each term.

$$\left(2m-3\right)\left(-m-\left(-1\right)\right)-3+1=0$$

The opposite of -1 is 1.

$$\left(2m-3\right)\left(-m+1\right)-3+1=0$$

Apply the distributive property by multiplying each term of 2m-3 by each term of -m+1.

$$-2m^{2}+2m+3m-3-3+1=0$$

Combine 2m and 3m to get 5m.

$$-2m^{2}+5m-3-3+1=0$$

Subtract 3 from -3 to get -6.

$$-2m^{2}+5m-6+1=0$$

Add -6 and 1 to get -5.

$$-2m^{2}+5m-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 5 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\left(-5\right)}}{2\left(-2\right)}$$

Square 5.

$$m=\frac{-5±\sqrt{25-4\left(-2\right)\left(-5\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$m=\frac{-5±\sqrt{25+8\left(-5\right)}}{2\left(-2\right)}$$

Multiply 8 times -5.

$$m=\frac{-5±\sqrt{25-40}}{2\left(-2\right)}$$

Add 25 to -40.

$$m=\frac{-5±\sqrt{-15}}{2\left(-2\right)}$$

Take the square root of -15.

$$m=\frac{-5±\sqrt{15}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$m=\frac{-5±\sqrt{15}i}{-4}$$

Now solve the equation m=\frac{-5±\sqrt{15}i}{-4} when ± is plus. Add -5 to i\sqrt{15}.

$$m=\frac{-5+\sqrt{15}i}{-4}$$

Divide -5+i\sqrt{15} by -4.

$$m=\frac{-\sqrt{15}i+5}{4}$$

Now solve the equation m=\frac{-5±\sqrt{15}i}{-4} when ± is minus. Subtract i\sqrt{15} from -5.

$$m=\frac{-\sqrt{15}i-5}{-4}$$

Divide -5-i\sqrt{15} by -4.

$$m=\frac{5+\sqrt{15}i}{4}$$

The equation is now solved.

$$m=\frac{-\sqrt{15}i+5}{4} m=\frac{5+\sqrt{15}i}{4}$$

Steps for Completing the Square

To find the opposite of m-1, find the opposite of each term.

$$\left(2m-3\right)\left(-m-\left(-1\right)\right)-3+1=0$$

The opposite of -1 is 1.

$$\left(2m-3\right)\left(-m+1\right)-3+1=0$$

Apply the distributive property by multiplying each term of 2m-3 by each term of -m+1.

$$-2m^{2}+2m+3m-3-3+1=0$$

Combine 2m and 3m to get 5m.

$$-2m^{2}+5m-3-3+1=0$$

Subtract 3 from -3 to get -6.

$$-2m^{2}+5m-6+1=0$$

Add -6 and 1 to get -5.

$$-2m^{2}+5m-5=0$$

Add 5 to both sides. Anything plus zero gives itself.

$$-2m^{2}+5m=5$$

Divide both sides by -2.

$$\frac{-2m^{2}+5m}{-2}=\frac{5}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$m^{2}+\frac{5}{-2}m=\frac{5}{-2}$$

Divide 5 by -2.

$$m^{2}-\frac{5}{2}m=\frac{5}{-2}$$

Divide 5 by -2.

$$m^{2}-\frac{5}{2}m=-\frac{5}{2}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{2}m+\left(-\frac{5}{4}\right)^{2}=-\frac{5}{2}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=-\frac{5}{2}+\frac{25}{16}$$

Add -\frac{5}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=-\frac{15}{16}$$

Factor m^{2}-\frac{5}{2}m+\frac{25}{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}.

$$\left(m-\frac{5}{4}\right)^{2}=-\frac{15}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{15}{16}}$$

Simplify.

$$m-\frac{5}{4}=\frac{\sqrt{15}i}{4} m-\frac{5}{4}=-\frac{\sqrt{15}i}{4}$$

Add \frac{5}{4} to both sides of the equation.

$$m=\frac{5+\sqrt{15}i}{4} m=\frac{-\sqrt{15}i+5}{4}$$