Solve 100/z-100/z-6=4/7

$$\frac { 100 } { z } - \frac { 100 } { z - 6 } = \frac { 4 } { 7 }$$

Solve for z

$$z=-\sqrt{1041}i+3\approx 3-32.26453161i$$

$$z=3+\sqrt{1041}i\approx 3+32.26453161i$$

Steps Using the Quadratic Formula

Variable z cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 7z\left(z-6\right), the least common multiple of z,z-6,7.

$$\left(7z-42\right)\times 100-7z\times 100=4z\left(z-6\right)$$

Use the distributive property to multiply 7z-42 by 100.

$$700z-4200-7z\times 100=4z\left(z-6\right)$$

Multiply 7 and 100 to get 700.

$$700z-4200-700z=4z\left(z-6\right)$$

Use the distributive property to multiply 4z by z-6.

$$700z-4200-700z=4z^{2}-24z$$

Subtract 4z^{2} from both sides.

$$700z-4200-700z-4z^{2}=-24z$$

Add 24z to both sides.

$$700z-4200-700z-4z^{2}+24z=0$$

Combine 700z and 24z to get 724z.

$$724z-4200-700z-4z^{2}=0$$

Combine 724z and -700z to get 24z.

$$24z-4200-4z^{2}=0$$

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.

$$-4z^{2}+24z-4200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 24 for b, and -4200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-24±\sqrt{24^{2}-4\left(-4\right)\left(-4200\right)}}{2\left(-4\right)}$$

Square 24.

$$z=\frac{-24±\sqrt{576-4\left(-4\right)\left(-4200\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$z=\frac{-24±\sqrt{576+16\left(-4200\right)}}{2\left(-4\right)}$$

Multiply 16 times -4200.

$$z=\frac{-24±\sqrt{576-67200}}{2\left(-4\right)}$$

Add 576 to -67200.

$$z=\frac{-24±\sqrt{-66624}}{2\left(-4\right)}$$

Take the square root of -66624.

$$z=\frac{-24±8\sqrt{1041}i}{2\left(-4\right)}$$

Multiply 2 times -4.

$$z=\frac{-24±8\sqrt{1041}i}{-8}$$

Now solve the equation z=\frac{-24±8\sqrt{1041}i}{-8} when ± is plus. Add -24 to 8i\sqrt{1041}.

$$z=\frac{-24+8\sqrt{1041}i}{-8}$$

Divide -24+8i\sqrt{1041} by -8.

$$z=-\sqrt{1041}i+3$$

Now solve the equation z=\frac{-24±8\sqrt{1041}i}{-8} when ± is minus. Subtract 8i\sqrt{1041} from -24.

$$z=\frac{-8\sqrt{1041}i-24}{-8}$$

Divide -24-8i\sqrt{1041} by -8.

$$z=3+\sqrt{1041}i$$

The equation is now solved.

$$z=-\sqrt{1041}i+3 z=3+\sqrt{1041}i$$

Steps for Completing the Square

Variable z cannot be equal to any of the values 0,6 since division by zero is not defined. Multiply both sides of the equation by 7z\left(z-6\right), the least common multiple of z,z-6,7.

$$\left(7z-42\right)\times 100-7z\times 100=4z\left(z-6\right)$$

Use the distributive property to multiply 7z-42 by 100.

$$700z-4200-7z\times 100=4z\left(z-6\right)$$

Multiply 7 and 100 to get 700.

$$700z-4200-700z=4z\left(z-6\right)$$

Use the distributive property to multiply 4z by z-6.

$$700z-4200-700z=4z^{2}-24z$$

Subtract 4z^{2} from both sides.

$$700z-4200-700z-4z^{2}=-24z$$

Add 24z to both sides.

$$700z-4200-700z-4z^{2}+24z=0$$

Combine 700z and 24z to get 724z.

$$724z-4200-700z-4z^{2}=0$$

Add 4200 to both sides. Anything plus zero gives itself.

$$724z-700z-4z^{2}=4200$$

Combine 724z and -700z to get 24z.

$$24z-4z^{2}=4200$$

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.

$$-4z^{2}+24z=4200$$

Divide both sides by -4.

$$\frac{-4z^{2}+24z}{-4}=\frac{4200}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$z^{2}+\frac{24}{-4}z=\frac{4200}{-4}$$

Divide 24 by -4.

$$z^{2}-6z=\frac{4200}{-4}$$

Divide 4200 by -4.

$$z^{2}-6z=-1050$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-6z+\left(-3\right)^{2}=-1050+\left(-3\right)^{2}$$

Square -3.

$$z^{2}-6z+9=-1050+9$$

Add -1050 to 9.

$$z^{2}-6z+9=-1041$$

Factor z^{2}-6z+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-3\right)^{2}=-1041$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-3\right)^{2}}=\sqrt{-1041}$$

Simplify.

$$z-3=\sqrt{1041}i z-3=-\sqrt{1041}i$$

Add 3 to both sides of the equation.

$$z=3+\sqrt{1041}i z=-\sqrt{1041}i+3$$