Solve quadtext{(2)}2m^2=5m-5

$$\quad \text { (2) } 2 m ^ { 2 } = 5 m - 5$$

Solve for m

$$m=\frac{5+\sqrt{55}i}{8}\approx 0.625+0.927024811i$$

$$m=\frac{-\sqrt{55}i+5}{8}\approx 0.625-0.927024811i$$

Steps Using the Quadratic Formula

Multiply 2 and 2 to get 4.

$$4m^{2}=5m-5$$

Subtract 5m from both sides.

$$4m^{2}-5m=-5$$

Add 5 to both sides.

$$4m^{2}-5m+5=0$$

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

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

Square -5.

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

Multiply -4 times 4.

$$m=\frac{-\left(-5\right)±\sqrt{25-16\times 5}}{2\times 4}$$

Multiply -16 times 5.

$$m=\frac{-\left(-5\right)±\sqrt{25-80}}{2\times 4}$$

Add 25 to -80.

$$m=\frac{-\left(-5\right)±\sqrt{-55}}{2\times 4}$$

Take the square root of -55.

$$m=\frac{-\left(-5\right)±\sqrt{55}i}{2\times 4}$$

The opposite of -5 is 5.

$$m=\frac{5±\sqrt{55}i}{2\times 4}$$

Multiply 2 times 4.

$$m=\frac{5±\sqrt{55}i}{8}$$

Now solve the equation m=\frac{5±\sqrt{55}i}{8} when ± is plus. Add 5 to i\sqrt{55}.

$$m=\frac{5+\sqrt{55}i}{8}$$

Now solve the equation m=\frac{5±\sqrt{55}i}{8} when ± is minus. Subtract i\sqrt{55} from 5.

$$m=\frac{-\sqrt{55}i+5}{8}$$

The equation is now solved.

$$m=\frac{5+\sqrt{55}i}{8} m=\frac{-\sqrt{55}i+5}{8}$$

Steps for Completing the Square

Multiply 2 and 2 to get 4.

$$4m^{2}=5m-5$$

Subtract 5m from both sides.

$$4m^{2}-5m=-5$$

Divide both sides by 4.

$$\frac{4m^{2}-5m}{4}=-\frac{5}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$m^{2}-\frac{5}{4}m=-\frac{5}{4}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{4}m+\left(-\frac{5}{8}\right)^{2}=-\frac{5}{4}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{4}m+\frac{25}{64}=-\frac{5}{4}+\frac{25}{64}$$

Add -\frac{5}{4} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{5}{4}m+\frac{25}{64}=-\frac{55}{64}$$

Factor m^{2}-\frac{5}{4}m+\frac{25}{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}.

$$\left(m-\frac{5}{8}\right)^{2}=-\frac{55}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{55}{64}}$$

Simplify.

$$m-\frac{5}{8}=\frac{\sqrt{55}i}{8} m-\frac{5}{8}=-\frac{\sqrt{55}i}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$m=\frac{5+\sqrt{55}i}{8} m=\frac{-\sqrt{55}i+5}{8}$$