Let `alpha in R` and let `alpha,beta` be the roots of the equation `x^(2)+60^(1/4)x+alpha=0`
If `alpha^(4)+beta^(4)=-30`,then the product of all possible values of `alpha` is

Answer

To solve the problem step-by-step, we start with the given quadratic equation and the conditions provided.

### Step 1: Identify the coefficients from the quadratic equation
The quadratic equation given is:
\[ x^2 + 60^{1/4} x + \alpha = 0 \]
From this, we can identify:
- \( a = 1 \)
- \( b = 60^{1/4} \)
- \( c = \alpha \)

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( ax^2 + bx + c = 0 \):
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} = -60^{1/4} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} = \alpha \)

### Step 3: Express \( \alpha^4 + \beta^4 \) in terms of \( \alpha \) and \( \beta \)
We know that:
\[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2\alpha^2\beta^2 \]
We can express \( \alpha^2 + \beta^2 \) using the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Vieta's formulas:
\[ \alpha^2 + \beta^2 = (-60^{1/4})^2 - 2\alpha = 60^{1/2} - 2\alpha \]

### Step 4: Substitute into the expression for \( \alpha^4 + \beta^4 \)
Now substituting \( \alpha^2 + \beta^2 \) into the equation for \( \alpha^4 + \beta^4 \):
\[ \alpha^4 + \beta^4 = (60^{1/2} - 2\alpha)^2 - 2\alpha^2 \]
Expanding this:
\[ = 60 - 4\alpha \cdot 60^{1/2} + 4\alpha^2 - 2\alpha^2 \]
\[ = 60 - 4\alpha \cdot 60^{1/2} + 2\alpha^2 \]

### Step 5: Set the equation equal to -30
Given that \( \alpha^4 + \beta^4 = -30 \), we set up the equation:
\[ 60 - 4\alpha \cdot 60^{1/2} + 2\alpha^2 = -30 \]
Rearranging gives:
\[ 2\alpha^2 - 4\alpha \cdot 60^{1/2} + 90 = 0 \]

### Step 6: Simplify the quadratic equation
Dividing the entire equation by 2:
\[ \alpha^2 - 2\alpha \cdot 60^{1/2} + 45 = 0 \]

### Step 7: Find the product of the roots
The product of the roots of the quadratic equation \( \alpha^2 + b\alpha + c = 0 \) is given by \( \frac{c}{a} \). Here:
- \( c = 45 \)
- \( a = 1 \)

Thus, the product of all possible values of \( \alpha \) is:
\[ \text{Product of roots} = 45 \]

### Final Answer
The product of all possible values of \( \alpha \) is \( 45 \).

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