QuestionFind all real solutions for the equation $z + \frac{16}{z+2} = 6$ . Enter answers as a comma-separated list.

Studdy Solution

STEP 1

Assumptions1. The equation is $z + \frac{16}{z+} =6$ . We are looking for real solutions

STEP 2

First, we need to eliminate the fraction in the equation. We can do this by multiplying the entire equation by $(z+2)$ .
$(z+2)(z + \frac{16}{z+2}) =6(z+2)$

STEP 3

Now, simplify the left side of the equation.
$z(z+2) +16 =6(z+2)$

STEP 4

Expand the terms on both sides of the equation.
$z^2 +2z +16 =6z +12$

STEP 5

Rearrange the equation to bring all terms to one side, resulting in a quadratic equation.
$z^2 +2z -z +16 -12 =0$

STEP 6

implify the equation.
$z^2 -4z +4 =0$

STEP 7

This is a quadratic equation in the form $ax^2 + bx + c =0$ . We can solve it using the quadratic formula $z = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .

STEP 8

Plug in the values for $a$ , $b$ , and $c$ from the equation $z^2 -4z +4 =0$ into the quadratic formula.
$z = \frac{-(-4) \pm \sqrt{(-4)^2 -414}}{2*1}$

STEP 9

implify the equation.
$z = \frac{4 \pm \sqrt{16 -16}}{2}$

STEP 10

Calculate the value under the square root.
$z = \frac{4 \pm \sqrt{0}}{2}$

STEP 11

implify the equation.
$z = \frac{4 \pm0}{}$

STEP 12

Calculate the value of $z$ .
$z = \frac{4}{2} =2$ The real solution of the equation is $z =2$ .

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