Solve 1-3z+275z^2-075z^3=0

$$1-3z+275 { z }^{ 2 } -075 { z }^{ 3 } = 0$$

Solve for z

$$z=\frac{3+\sqrt{1091}i}{550}\approx 0.005454545+0.060055071i$$

$$z=\frac{-\sqrt{1091}i+3}{550}\approx 0.005454545-0.060055071i$$

Steps Using the Quadratic Formula

Multiply 0 and 75 to get 0.

$$1-3z+275z^{2}-0z^{3}=0$$

Anything times zero gives zero.

$$1-3z+275z^{2}-0=0$$

Reorder the terms.

$$275z^{2}-3z+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 275 for a, -3 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 275}}{2\times 275}$$

Square -3.

$$z=\frac{-\left(-3\right)±\sqrt{9-4\times 275}}{2\times 275}$$

Multiply -4 times 275.

$$z=\frac{-\left(-3\right)±\sqrt{9-1100}}{2\times 275}$$

Add 9 to -1100.

$$z=\frac{-\left(-3\right)±\sqrt{-1091}}{2\times 275}$$

Take the square root of -1091.

$$z=\frac{-\left(-3\right)±\sqrt{1091}i}{2\times 275}$$

The opposite of -3 is 3.

$$z=\frac{3±\sqrt{1091}i}{2\times 275}$$

Multiply 2 times 275.

$$z=\frac{3±\sqrt{1091}i}{550}$$

Now solve the equation z=\frac{3±\sqrt{1091}i}{550} when ± is plus. Add 3 to i\sqrt{1091}.

$$z=\frac{3+\sqrt{1091}i}{550}$$

Now solve the equation z=\frac{3±\sqrt{1091}i}{550} when ± is minus. Subtract i\sqrt{1091} from 3.

$$z=\frac{-\sqrt{1091}i+3}{550}$$

The equation is now solved.

$$z=\frac{3+\sqrt{1091}i}{550} z=\frac{-\sqrt{1091}i+3}{550}$$

Steps for Completing the Square

Multiply 0 and 75 to get 0.

$$1-3z+275z^{2}-0z^{3}=0$$

Anything times zero gives zero.

$$1-3z+275z^{2}-0=0$$

Add 0 to both sides.

$$1-3z+275z^{2}=0+0$$

Add 0 and 0 to get 0.

$$1-3z+275z^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-3z+275z^{2}=-1$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$275z^{2}-3z=-1$$

Divide both sides by 275.

$$\frac{275z^{2}-3z}{275}=-\frac{1}{275}$$

Dividing by 275 undoes the multiplication by 275.

$$z^{2}-\frac{3}{275}z=-\frac{1}{275}$$

Divide -\frac{3}{275}, the coefficient of the x term, by 2 to get -\frac{3}{550}. Then add the square of -\frac{3}{550} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{3}{275}z+\left(-\frac{3}{550}\right)^{2}=-\frac{1}{275}+\left(-\frac{3}{550}\right)^{2}$$

Square -\frac{3}{550} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{3}{275}z+\frac{9}{302500}=-\frac{1}{275}+\frac{9}{302500}$$

Add -\frac{1}{275} to \frac{9}{302500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}-\frac{3}{275}z+\frac{9}{302500}=-\frac{1091}{302500}$$

Factor z^{2}-\frac{3}{275}z+\frac{9}{302500}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{3}{550}\right)^{2}=-\frac{1091}{302500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{3}{550}\right)^{2}}=\sqrt{-\frac{1091}{302500}}$$

Simplify.

$$z-\frac{3}{550}=\frac{\sqrt{1091}i}{550} z-\frac{3}{550}=-\frac{\sqrt{1091}i}{550}$$

Add \frac{3}{550} to both sides of the equation.

$$z=\frac{3+\sqrt{1091}i}{550} z=\frac{-\sqrt{1091}i+3}{550}$$