Solve 8/21-3/7z=z/3

$$\frac { 8 } { 21 } - \frac { 3 } { 7 z } = \frac { z } { 3 }$$

Solve for z

$$z=\frac{-\sqrt{47}i+4}{7}\approx 0.571428571-0.979379229i$$

$$z=\frac{4+\sqrt{47}i}{7}\approx 0.571428571+0.979379229i$$

Steps Using the Quadratic Formula

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 21z, the least common multiple of 21,7z,3.

$$21z\times \frac{8}{21}-3\times 3=7zz$$

Multiply 21 and \frac{8}{21} to get 8.

$$8z-3\times 3=7zz$$

Multiply -3 and 3 to get -9.

$$8z-9=7zz$$

Multiply z and z to get z^{2}.

$$8z-9=7z^{2}$$

Subtract 7z^{2} from both sides.

$$8z-9-7z^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-7z^{2}+8z-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-8±\sqrt{8^{2}-4\left(-7\right)\left(-9\right)}}{2\left(-7\right)}$$

Square 8.

$$z=\frac{-8±\sqrt{64-4\left(-7\right)\left(-9\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$z=\frac{-8±\sqrt{64+28\left(-9\right)}}{2\left(-7\right)}$$

Multiply 28 times -9.

$$z=\frac{-8±\sqrt{64-252}}{2\left(-7\right)}$$

Add 64 to -252.

$$z=\frac{-8±\sqrt{-188}}{2\left(-7\right)}$$

Take the square root of -188.

$$z=\frac{-8±2\sqrt{47}i}{2\left(-7\right)}$$

Multiply 2 times -7.

$$z=\frac{-8±2\sqrt{47}i}{-14}$$

Now solve the equation z=\frac{-8±2\sqrt{47}i}{-14} when ± is plus. Add -8 to 2i\sqrt{47}.

$$z=\frac{-8+2\sqrt{47}i}{-14}$$

Divide -8+2i\sqrt{47} by -14.

$$z=\frac{-\sqrt{47}i+4}{7}$$

Now solve the equation z=\frac{-8±2\sqrt{47}i}{-14} when ± is minus. Subtract 2i\sqrt{47} from -8.

$$z=\frac{-2\sqrt{47}i-8}{-14}$$

Divide -8-2i\sqrt{47} by -14.

$$z=\frac{4+\sqrt{47}i}{7}$$

The equation is now solved.

$$z=\frac{-\sqrt{47}i+4}{7} z=\frac{4+\sqrt{47}i}{7}$$

Steps for Completing the Square

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 21z, the least common multiple of 21,7z,3.

$$21z\times \frac{8}{21}-3\times 3=7zz$$

Multiply 21 and \frac{8}{21} to get 8.

$$8z-3\times 3=7zz$$

Multiply -3 and 3 to get -9.

$$8z-9=7zz$$

Multiply z and z to get z^{2}.

$$8z-9=7z^{2}$$

Subtract 7z^{2} from both sides.

$$8z-9-7z^{2}=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$8z-7z^{2}=9$$

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.

$$-7z^{2}+8z=9$$

Divide both sides by -7.

$$\frac{-7z^{2}+8z}{-7}=\frac{9}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$z^{2}+\frac{8}{-7}z=\frac{9}{-7}$$

Divide 8 by -7.

$$z^{2}-\frac{8}{7}z=\frac{9}{-7}$$

Divide 9 by -7.

$$z^{2}-\frac{8}{7}z=-\frac{9}{7}$$

Divide -\frac{8}{7}, the coefficient of the x term, by 2 to get -\frac{4}{7}. Then add the square of -\frac{4}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{8}{7}z+\left(-\frac{4}{7}\right)^{2}=-\frac{9}{7}+\left(-\frac{4}{7}\right)^{2}$$

Square -\frac{4}{7} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{8}{7}z+\frac{16}{49}=-\frac{9}{7}+\frac{16}{49}$$

Add -\frac{9}{7} to \frac{16}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}-\frac{8}{7}z+\frac{16}{49}=-\frac{47}{49}$$

Factor z^{2}-\frac{8}{7}z+\frac{16}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{4}{7}\right)^{2}=-\frac{47}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{4}{7}\right)^{2}}=\sqrt{-\frac{47}{49}}$$

Simplify.

$$z-\frac{4}{7}=\frac{\sqrt{47}i}{7} z-\frac{4}{7}=-\frac{\sqrt{47}i}{7}$$

Add \frac{4}{7} to both sides of the equation.

$$z=\frac{4+\sqrt{47}i}{7} z=\frac{-\sqrt{47}i+4}{7}$$