Solve z^2-1.386z+0.368+0.386z+0.264=0

$$z ^ { 2 } - 1.386 z + 0.368 + 0.386 z + 0.264 = 0$$

Solve for z

$$z=\frac{i\sqrt{955}}{50}+0.5\approx 0.5+0.618061486i$$

$$z=-\frac{i\sqrt{955}}{50}+0.5\approx 0.5-0.618061486i$$

Steps Using the Quadratic Formula

Combine -1.386z and 0.386z to get -z.

$$z^{2}-z+0.368+0.264=0$$

Add 0.368 and 0.264 to get 0.632.

$$z^{2}-z+0.632=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 0.632 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-1\right)±\sqrt{1-4\times 0.632}}{2}$$

Multiply -4 times 0.632.

$$z=\frac{-\left(-1\right)±\sqrt{1-2.528}}{2}$$

Add 1 to -2.528.

$$z=\frac{-\left(-1\right)±\sqrt{-1.528}}{2}$$

Take the square root of -1.528.

$$z=\frac{-\left(-1\right)±\frac{\sqrt{955}i}{25}}{2}$$

The opposite of -1 is 1.

$$z=\frac{1±\frac{\sqrt{955}i}{25}}{2}$$

Now solve the equation z=\frac{1±\frac{\sqrt{955}i}{25}}{2} when ± is plus. Add 1 to \frac{i\sqrt{955}}{25}.

$$z=\frac{\frac{\sqrt{955}i}{25}+1}{2}$$

Divide 1+\frac{i\sqrt{955}}{25} by 2.

$$z=\frac{\sqrt{955}i}{50}+\frac{1}{2}$$

Now solve the equation z=\frac{1±\frac{\sqrt{955}i}{25}}{2} when ± is minus. Subtract \frac{i\sqrt{955}}{25} from 1.

$$z=\frac{-\frac{\sqrt{955}i}{25}+1}{2}$$

Divide 1-\frac{i\sqrt{955}}{25} by 2.

$$z=-\frac{\sqrt{955}i}{50}+\frac{1}{2}$$

The equation is now solved.

$$z=\frac{\sqrt{955}i}{50}+\frac{1}{2} z=-\frac{\sqrt{955}i}{50}+\frac{1}{2}$$

Steps for Completing the Square

Combine -1.386z and 0.386z to get -z.

$$z^{2}-z+0.368+0.264=0$$

Add 0.368 and 0.264 to get 0.632.

$$z^{2}-z+0.632=0$$

Subtract 0.632 from both sides. Anything subtracted from zero gives its negation.

$$z^{2}-z=-0.632$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-z+\left(-\frac{1}{2}\right)^{2}=-0.632+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-z+\frac{1}{4}=-0.632+\frac{1}{4}$$

Add -0.632 to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}-z+\frac{1}{4}=-\frac{191}{500}$$

Factor z^{2}-z+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{1}{2}\right)^{2}=-\frac{191}{500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{191}{500}}$$

Simplify.

$$z-\frac{1}{2}=\frac{\sqrt{955}i}{50} z-\frac{1}{2}=-\frac{\sqrt{955}i}{50}$$

Add \frac{1}{2} to both sides of the equation.

$$z=\frac{\sqrt{955}i}{50}+\frac{1}{2} z=-\frac{\sqrt{955}i}{50}+\frac{1}{2}$$