If `alpha " & " beta` are the roots of equation `ax^(2) + bx + c= 0` then find the quadratic equation whose roots are `alpha^(2) " & " beta^(2)`

Answer

To find the quadratic equation whose roots are \( \alpha^2 \) and \( \beta^2 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Identify the sum and product of the roots \( \alpha \) and \( \beta \)
From Vieta's formulas:
- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \beta = \frac{c}{a} \)

### Step 2: Find the sum of the new roots \( \alpha^2 \) and \( \beta^2 \)
We can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values we found:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \]
Calculating this gives:
\[ \alpha^2 + \beta^2 = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2} \]

### Step 3: Find the product of the new roots \( \alpha^2 \) and \( \beta^2 \)
The product of the new roots can be expressed as:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 \]
Substituting the value we found:
\[ \alpha^2 \beta^2 = \left(\frac{c}{a}\right)^2 = \frac{c^2}{a^2} \]

### Step 4: Form the new quadratic equation
Using the sum and product of the new roots, we can write the quadratic equation in the standard form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we calculated:
\[ x^2 - \left(\frac{b^2 - 2ac}{a^2}\right)x + \left(\frac{c^2}{a^2}\right) = 0 \]

### Step 5: Clear the denominators
To eliminate the denominators, multiply the entire equation by \( a^2 \):
\[ a^2 x^2 - (b^2 - 2ac)x + c^2 = 0 \]

Thus, the required quadratic equation whose roots are \( \alpha^2 \) and \( \beta^2 \) is:
\[ a^2 x^2 - (b^2 - 2ac)x + c^2 = 0 \]