Solve 2/z-1-5/4=-1/z+1

$$\frac { 2 } { z - 1 } - \frac { 5 } { 4 } = \frac { - 1 } { z + 1 }$$

Solve for z

$$z=-\frac{3}{5}=-0.6$$

$$z=3$$

Steps Using the Quadratic Formula

Variable z cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 4\left(z-1\right)\left(z+1\right), the least common multiple of z-1,4,z+1.

$$\left(4z+4\right)\times 2+4\left(z-1\right)\left(z+1\right)\left(-\frac{5}{4}\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply 4z+4 by 2.

$$8z+8+4\left(z-1\right)\left(z+1\right)\left(-\frac{5}{4}\right)=\left(4z-4\right)\left(-1\right)$$

Multiply 4 and -\frac{5}{4} to get -5.

$$8z+8-5\left(z-1\right)\left(z+1\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply -5 by z-1.

$$8z+8+\left(-5z+5\right)\left(z+1\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply -5z+5 by z+1 and combine like terms.

$$8z+8-5z^{2}+5=\left(4z-4\right)\left(-1\right)$$

Add 8 and 5 to get 13.

$$8z+13-5z^{2}=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply 4z-4 by -1.

$$8z+13-5z^{2}=-4z+4$$

Add 4z to both sides.

$$8z+13-5z^{2}+4z=4$$

Combine 8z and 4z to get 12z.

$$12z+13-5z^{2}=4$$

Subtract 4 from both sides.

$$12z+13-5z^{2}-4=0$$

Subtract 4 from 13 to get 9.

$$12z+9-5z^{2}=0$$

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.

$$-5z^{2}+12z+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 12 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-12±\sqrt{12^{2}-4\left(-5\right)\times 9}}{2\left(-5\right)}$$

Square 12.

$$z=\frac{-12±\sqrt{144-4\left(-5\right)\times 9}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$z=\frac{-12±\sqrt{144+20\times 9}}{2\left(-5\right)}$$

Multiply 20 times 9.

$$z=\frac{-12±\sqrt{144+180}}{2\left(-5\right)}$$

Add 144 to 180.

$$z=\frac{-12±\sqrt{324}}{2\left(-5\right)}$$

Take the square root of 324.

$$z=\frac{-12±18}{2\left(-5\right)}$$

Multiply 2 times -5.

$$z=\frac{-12±18}{-10}$$

Now solve the equation z=\frac{-12±18}{-10} when ± is plus. Add -12 to 18.

$$z=\frac{6}{-10}$$

Reduce the fraction \frac{6}{-10} to lowest terms by extracting and canceling out 2.

$$z=-\frac{3}{5}$$

Now solve the equation z=\frac{-12±18}{-10} when ± is minus. Subtract 18 from -12.

$$z=-\frac{30}{-10}$$

Divide -30 by -10.

$$z=3$$

The equation is now solved.

$$z=-\frac{3}{5} z=3$$

Steps for Completing the Square

Variable z cannot be equal to any of the values -1,1 since division by zero is not defined. Multiply both sides of the equation by 4\left(z-1\right)\left(z+1\right), the least common multiple of z-1,4,z+1.

$$\left(4z+4\right)\times 2+4\left(z-1\right)\left(z+1\right)\left(-\frac{5}{4}\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply 4z+4 by 2.

$$8z+8+4\left(z-1\right)\left(z+1\right)\left(-\frac{5}{4}\right)=\left(4z-4\right)\left(-1\right)$$

Multiply 4 and -\frac{5}{4} to get -5.

$$8z+8-5\left(z-1\right)\left(z+1\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply -5 by z-1.

$$8z+8+\left(-5z+5\right)\left(z+1\right)=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply -5z+5 by z+1 and combine like terms.

$$8z+8-5z^{2}+5=\left(4z-4\right)\left(-1\right)$$

Add 8 and 5 to get 13.

$$8z+13-5z^{2}=\left(4z-4\right)\left(-1\right)$$

Use the distributive property to multiply 4z-4 by -1.

$$8z+13-5z^{2}=-4z+4$$

Add 4z to both sides.

$$8z+13-5z^{2}+4z=4$$

Combine 8z and 4z to get 12z.

$$12z+13-5z^{2}=4$$

Subtract 13 from both sides.

$$12z-5z^{2}=4-13$$

Subtract 13 from 4 to get -9.

$$12z-5z^{2}=-9$$

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.

$$-5z^{2}+12z=-9$$

Divide both sides by -5.

$$\frac{-5z^{2}+12z}{-5}=-\frac{9}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$z^{2}+\frac{12}{-5}z=-\frac{9}{-5}$$

Divide 12 by -5.

$$z^{2}-\frac{12}{5}z=-\frac{9}{-5}$$

Divide -9 by -5.

$$z^{2}-\frac{12}{5}z=\frac{9}{5}$$

Divide -\frac{12}{5}, the coefficient of the x term, by 2 to get -\frac{6}{5}. Then add the square of -\frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{12}{5}z+\left(-\frac{6}{5}\right)^{2}=\frac{9}{5}+\left(-\frac{6}{5}\right)^{2}$$

Square -\frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{12}{5}z+\frac{36}{25}=\frac{9}{5}+\frac{36}{25}$$

Add \frac{9}{5} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}-\frac{12}{5}z+\frac{36}{25}=\frac{81}{25}$$

Factor z^{2}-\frac{12}{5}z+\frac{36}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{6}{5}\right)^{2}=\frac{81}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{6}{5}\right)^{2}}=\sqrt{\frac{81}{25}}$$

Simplify.

$$z-\frac{6}{5}=\frac{9}{5} z-\frac{6}{5}=-\frac{9}{5}$$

Add \frac{6}{5} to both sides of the equation.

$$z=3 z=-\frac{3}{5}$$