If the roots of a quadratic equation `ax^(2)+ bx +c = 0` are `alpha and beta` then the quadratic equation having roots `alpha^(2) and beta^(2)` is :

Options

A. `x^(2) -(b^(2) - 2ac) x + c = 0`
B. `a^(2) x^(2) - (b^(2) - 2ac) x + c = 0`
C. `ax^(2) - (b^(2) -2ac) x + c^(2) = 0`
D. `a^(2) x^(2) - (b^(2) - 2ac) x + c^(2) =0`

Answer

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

### Step 1: Find the sum of the roots \( \alpha \) and \( \beta \)
The sum of the roots \( \alpha + \beta \) can be expressed using the coefficients of the original quadratic equation:
\[ \alpha + \beta = -\frac{b}{a} \]

### Step 2: Find the product of the roots \( \alpha \) and \( \beta \)
The product of the roots \( \alpha \beta \) can also be expressed using the coefficients:
\[ \alpha \beta = \frac{c}{a} \]

### Step 3: Find the sum of the new roots \( \alpha^2 \) and \( \beta^2 \)
Using the identity for the sum of squares, we have:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from Steps 1 and 2:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) \]
\[ = \frac{b^2}{a^2} - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2} \]

### Step 4: Find the product of the new roots \( \alpha^2 \) and \( \beta^2 \)
The product of the new roots can be calculated as:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 = \left(\frac{c}{a}\right)^2 = \frac{c^2}{a^2} \]

### Step 5: Form the new quadratic equation
The quadratic equation with roots \( \alpha^2 \) and \( \beta^2 \) can be expressed in the standard form:
\[ x^2 - (\text{sum of the roots})x + (\text{product of the roots}) = 0 \]
Substituting the values from Steps 3 and 4:
\[ x^2 - \left(\frac{b^2 - 2ac}{a^2}\right)x + \left(\frac{c^2}{a^2}\right) = 0 \]

### Step 6: Clear the denominators
To eliminate the fractions, multiply the entire equation by \( a^2 \):
\[ a^2 x^2 - (b^2 - 2ac)x + c^2 = 0 \]

### Final Result
Thus, the quadratic equation having roots \( \alpha^2 \) and \( \beta^2 \) is:
\[ a^2 x^2 - (b^2 - 2ac)x + c^2 = 0 \]