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To solve the equation [tex]\(\frac{y}{y-4} - \frac{4}{y+4} = \frac{32}{y^2 - 18}\)[/tex], we need to follow these steps:
### Step 1: Identify the Domain
First, identify values for which the expressions are undefined. The denominators [tex]\(y-4\)[/tex], [tex]\(y+4\)[/tex], and [tex]\(y^2 - 18\)[/tex] must not be zero.
- [tex]\(y-4=0 \implies y=4\)[/tex]
- [tex]\(y+4=0 \implies y=-4\)[/tex]
- [tex]\(y^2 - 18 = 0 \implies y^2 = 18 \implies y = \pm \sqrt{18} = \pm 3\sqrt{2}\)[/tex]
Therefore, [tex]\(y\)[/tex] cannot be [tex]\(4\)[/tex], [tex]\(-4\)[/tex], [tex]\(3\sqrt{2}\)[/tex], or [tex]\(-3\sqrt{2}\)[/tex].
### Step 2: Rewrite the Equation
Recognize that by rewriting the right-hand side [tex]\(\frac{32}{y^2 - 18}\)[/tex]:
[tex]\[ y^2 - 18 = (y - 3\sqrt{2})(y + 3\sqrt{2}) \][/tex]
This shows it's zero at [tex]\(\pm 3\sqrt{2}\)[/tex].
### Step 3: Simplify the Left-hand Side
To combine the terms on the left-hand side, we need a common denominator:
[tex]\[ \frac{y(y+4) - 4(y-4)}{(y-4)(y+4)} \][/tex]
[tex]\[ \frac{y^2 + 4y - 4y + 16}{y^2 - 16} \][/tex]
[tex]\[ \frac{y^2 + 16}{y^2 - 16} \][/tex]
### Step 4: Simplify the Entire Equation
We get the equation:
[tex]\[ \frac{y^2 + 16}{y^2 - 16} = \frac{32}{y^2 - 18} \][/tex]
Cross multiplying gives:
[tex]\[ (y^2 + 16)(y^2 - 18) = 32(y^2 - 16) \][/tex]
Expanding both sides:
[tex]\[ y^4 - 18y^2 + 16y^2 + 16(-18) = 32y^2 - 512 \][/tex]
[tex]\[ y^4 - 2y^2 - 288 = 32y^2 - 512 \][/tex]
[tex]\[ y^4 - 34y^2 + 224 = 0 \][/tex]
### Step 5: Solve the Polynomial Equation
Solve the equation:
[tex]\[ y^4 - 34y^2 + 224 = 0 \][/tex]
Let [tex]\(u = y^2\)[/tex], then we have a quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[ u^2 - 34u + 224 = 0 \][/tex]
Using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\(a=1\)[/tex], [tex]\(b=-34\)[/tex], and [tex]\(c=224\)[/tex]:
[tex]\[ u = \frac{34 \pm \sqrt{1156 - 896}}{2} = \frac{34 \pm \sqrt{260}}{2} = \frac{34 \pm 2\sqrt{65}}{2} \][/tex]
[tex]\[ u = 17 \pm \sqrt{65} \][/tex]
Thus:
[tex]\[ y^2 = 17 \pm \sqrt{65} \][/tex]
Taking the square root of both sides, we find four possible solutions for [tex]\(y\)[/tex]:
[tex]\[ y = \pm \sqrt{17 - \sqrt{65}} \][/tex]
[tex]\[ y = \pm \sqrt{17 + \sqrt{65}} \][/tex]
### Step 6: Check for Prohibited Values and Validity
Check if these solutions are in the domain and exclude prohibited values:
[tex]\[ \pm \sqrt{17 - \sqrt{65}}, \pm \sqrt{17 + \sqrt{65}} \][/tex]
These values are real and not equal to [tex]\(\pm 4\)[/tex] or [tex]\( \pm 3\sqrt{2}\)[/tex].
Thus, the solutions are:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
### Conclusion
In conclusion, the solution to the equation [tex]\(\frac{y}{y-4} - \frac{4}{y+4} = \frac{32}{y^2 - 18}\)[/tex] are:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
These values are not [tex]\(0\)[/tex]. Hence, checking our initial options:
- [tex]\( y = -4\)[/tex] and [tex]\(y = 4\)[/tex] are not solutions.
- [tex]\( y = 0\)[/tex] is not a solution.
- All real numbers are not solutions.
- There is a solution.
Therefore, the answer is none of the options provided directly and the explicit solutions are given as:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
### Step 1: Identify the Domain
First, identify values for which the expressions are undefined. The denominators [tex]\(y-4\)[/tex], [tex]\(y+4\)[/tex], and [tex]\(y^2 - 18\)[/tex] must not be zero.
- [tex]\(y-4=0 \implies y=4\)[/tex]
- [tex]\(y+4=0 \implies y=-4\)[/tex]
- [tex]\(y^2 - 18 = 0 \implies y^2 = 18 \implies y = \pm \sqrt{18} = \pm 3\sqrt{2}\)[/tex]
Therefore, [tex]\(y\)[/tex] cannot be [tex]\(4\)[/tex], [tex]\(-4\)[/tex], [tex]\(3\sqrt{2}\)[/tex], or [tex]\(-3\sqrt{2}\)[/tex].
### Step 2: Rewrite the Equation
Recognize that by rewriting the right-hand side [tex]\(\frac{32}{y^2 - 18}\)[/tex]:
[tex]\[ y^2 - 18 = (y - 3\sqrt{2})(y + 3\sqrt{2}) \][/tex]
This shows it's zero at [tex]\(\pm 3\sqrt{2}\)[/tex].
### Step 3: Simplify the Left-hand Side
To combine the terms on the left-hand side, we need a common denominator:
[tex]\[ \frac{y(y+4) - 4(y-4)}{(y-4)(y+4)} \][/tex]
[tex]\[ \frac{y^2 + 4y - 4y + 16}{y^2 - 16} \][/tex]
[tex]\[ \frac{y^2 + 16}{y^2 - 16} \][/tex]
### Step 4: Simplify the Entire Equation
We get the equation:
[tex]\[ \frac{y^2 + 16}{y^2 - 16} = \frac{32}{y^2 - 18} \][/tex]
Cross multiplying gives:
[tex]\[ (y^2 + 16)(y^2 - 18) = 32(y^2 - 16) \][/tex]
Expanding both sides:
[tex]\[ y^4 - 18y^2 + 16y^2 + 16(-18) = 32y^2 - 512 \][/tex]
[tex]\[ y^4 - 2y^2 - 288 = 32y^2 - 512 \][/tex]
[tex]\[ y^4 - 34y^2 + 224 = 0 \][/tex]
### Step 5: Solve the Polynomial Equation
Solve the equation:
[tex]\[ y^4 - 34y^2 + 224 = 0 \][/tex]
Let [tex]\(u = y^2\)[/tex], then we have a quadratic equation in terms of [tex]\(u\)[/tex]:
[tex]\[ u^2 - 34u + 224 = 0 \][/tex]
Using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\(a=1\)[/tex], [tex]\(b=-34\)[/tex], and [tex]\(c=224\)[/tex]:
[tex]\[ u = \frac{34 \pm \sqrt{1156 - 896}}{2} = \frac{34 \pm \sqrt{260}}{2} = \frac{34 \pm 2\sqrt{65}}{2} \][/tex]
[tex]\[ u = 17 \pm \sqrt{65} \][/tex]
Thus:
[tex]\[ y^2 = 17 \pm \sqrt{65} \][/tex]
Taking the square root of both sides, we find four possible solutions for [tex]\(y\)[/tex]:
[tex]\[ y = \pm \sqrt{17 - \sqrt{65}} \][/tex]
[tex]\[ y = \pm \sqrt{17 + \sqrt{65}} \][/tex]
### Step 6: Check for Prohibited Values and Validity
Check if these solutions are in the domain and exclude prohibited values:
[tex]\[ \pm \sqrt{17 - \sqrt{65}}, \pm \sqrt{17 + \sqrt{65}} \][/tex]
These values are real and not equal to [tex]\(\pm 4\)[/tex] or [tex]\( \pm 3\sqrt{2}\)[/tex].
Thus, the solutions are:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
### Conclusion
In conclusion, the solution to the equation [tex]\(\frac{y}{y-4} - \frac{4}{y+4} = \frac{32}{y^2 - 18}\)[/tex] are:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
These values are not [tex]\(0\)[/tex]. Hence, checking our initial options:
- [tex]\( y = -4\)[/tex] and [tex]\(y = 4\)[/tex] are not solutions.
- [tex]\( y = 0\)[/tex] is not a solution.
- All real numbers are not solutions.
- There is a solution.
Therefore, the answer is none of the options provided directly and the explicit solutions are given as:
[tex]\[ y = -\sqrt{17 - \sqrt{65}}, \sqrt{17 - \sqrt{65}}, -\sqrt{17 + \sqrt{65}}, \sqrt{17 + \sqrt{65}} \][/tex]
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