Solve 4/z+z/4=(12-z/z)(12+z/4)

$$\frac { 4 } { z } + \frac { z } { 4 } = ( \frac { 12 - z } { z } ) ( \frac { 12 + z } { 4 } )$$

Solve for z

$$z=-8$$

$$z=8$$

Steps by Finding Square Root

Variable z cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4z, the least common multiple of z,4.

$$4\times 4+zz=4\left(12-z\right)\times \frac{12+z}{4}$$

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

$$4\times 4+z^{2}=4\left(12-z\right)\times \frac{12+z}{4}$$

Multiply 4 and 4 to get 16.

$$16+z^{2}=4\left(12-z\right)\times \frac{12+z}{4}$$

Express 4\times \frac{12+z}{4} as a single fraction.

$$16+z^{2}=\frac{4\left(12+z\right)}{4}\left(12-z\right)$$

Cancel out 4 and 4.

$$16+z^{2}=\left(12+z\right)\left(12-z\right)$$

Consider \left(12+z\right)\left(12-z\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16+z^{2}=12^{2}-z^{2}$$

Calculate 12 to the power of 2 and get 144.

$$16+z^{2}=144-z^{2}$$

Add z^{2} to both sides.

$$16+z^{2}+z^{2}=144$$

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

$$16+2z^{2}=144$$

Subtract 16 from both sides.

$$2z^{2}=144-16$$

Subtract 16 from 144 to get 128.

$$2z^{2}=128$$

Divide both sides by 2.

$$z^{2}=\frac{128}{2}$$

Divide 128 by 2 to get 64.

$$z^{2}=64$$

Take the square root of both sides of the equation.

$$z=8 z=-8$$

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 4z, the least common multiple of z,4.

$$4\times 4+zz=4\left(12-z\right)\times \frac{12+z}{4}$$

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

$$4\times 4+z^{2}=4\left(12-z\right)\times \frac{12+z}{4}$$

Multiply 4 and 4 to get 16.

$$16+z^{2}=4\left(12-z\right)\times \frac{12+z}{4}$$

Express 4\times \frac{12+z}{4} as a single fraction.

$$16+z^{2}=\frac{4\left(12+z\right)}{4}\left(12-z\right)$$

Cancel out 4 and 4.

$$16+z^{2}=\left(12+z\right)\left(12-z\right)$$

Consider \left(12+z\right)\left(12-z\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16+z^{2}=12^{2}-z^{2}$$

Calculate 12 to the power of 2 and get 144.

$$16+z^{2}=144-z^{2}$$

Subtract 144 from both sides.

$$16+z^{2}-144=-z^{2}$$

Subtract 144 from 16 to get -128.

$$-128+z^{2}=-z^{2}$$

Add z^{2} to both sides.

$$-128+z^{2}+z^{2}=0$$

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

$$-128+2z^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$2z^{2}-128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{0±\sqrt{0^{2}-4\times 2\left(-128\right)}}{2\times 2}$$

Square 0.

$$z=\frac{0±\sqrt{-4\times 2\left(-128\right)}}{2\times 2}$$

Multiply -4 times 2.

$$z=\frac{0±\sqrt{-8\left(-128\right)}}{2\times 2}$$

Multiply -8 times -128.

$$z=\frac{0±\sqrt{1024}}{2\times 2}$$

Take the square root of 1024.

$$z=\frac{0±32}{2\times 2}$$

Multiply 2 times 2.

$$z=\frac{0±32}{4}$$

Now solve the equation z=\frac{0±32}{4} when ± is plus. Divide 32 by 4.

$$z=8$$

Now solve the equation z=\frac{0±32}{4} when ± is minus. Divide -32 by 4.

$$z=-8$$

The equation is now solved.

$$z=8 z=-8$$