If `alpha and beta` are the roots of the equation `x^(2)+x-7=0`, form the equation whose roots are `alpha^(2)` and `beta^(2)`.

Answer

To find the quadratic equation whose roots are \( \alpha^2 \) and \( \beta^2 \), we start by analyzing the given quadratic equation:

### Step 1: Identify the coefficients
The given equation is:
\[ x^2 + x - 7 = 0 \]
Here, we can identify:
- \( a = 1 \)
- \( b = 1 \)
- \( c = -7 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{1}{1} = -1 \]
- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{-7}{1} = -7 \]

### Step 3: Calculate \( \alpha^2 + \beta^2 \)
To find \( \alpha^2 + \beta^2 \), we use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = (-1)^2 - 2(-7) = 1 + 14 = 15 \]

### Step 4: Calculate \( \alpha^2 \beta^2 \)
To find \( \alpha^2 \beta^2 \), we use the identity:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 \]
Substituting the value of \( \alpha \beta \):
\[ \alpha^2 \beta^2 = (-7)^2 = 49 \]

### Step 5: Form the new quadratic equation
The quadratic equation with roots \( \alpha^2 \) and \( \beta^2 \) can be written as:
\[ x^2 - (\alpha^2 + \beta^2)x + \alpha^2 \beta^2 = 0 \]
Substituting the values we calculated:
\[ x^2 - 15x + 49 = 0 \]

Thus, the required quadratic equation whose roots are \( \alpha^2 \) and \( \beta^2 \) is:
\[ \boxed{x^2 - 15x + 49 = 0} \]