If `alpha` and `beta` are the roots of the equation `2x^2-3x + 4=0`, then the equation whose roots are `alpha^2` and `beta^2,` is a. `4x^(2)+7x+16=0` b. `4x^(2)+7x+6=0` c. `4x^(2)+7x+1=0` d. `4x^(2)-7x+16=0`

Options

A. `4x^(2)+7x+16=0`
B. `4x^(2)+7x+6=0`
C. `4x^(2)+7x+1=0`
D. `4x^(2)-7x+16=0`

Answer

To solve the problem step by step, we will follow the method of finding the new quadratic equation whose roots are the squares of the roots of the given equation.

### Step 1: Identify the coefficients of the given quadratic equation
The given equation is:
\[ 2x^2 - 3x + 4 = 0 \]

Here, we have:
- \( a = 2 \)
- \( b = -3 \)
- \( c = 4 \)

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-3}{2} = \frac{3}{2} \]

- The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{4}{2} = 2 \]

### Step 3: Calculate the sum of the squares of the roots
To find \( \alpha^2 + \beta^2 \), we can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]

Substituting the values we found:
\[ \alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2 \cdot 2 \]
\[ = \frac{9}{4} - 4 \]
\[ = \frac{9}{4} - \frac{16}{4} \]
\[ = \frac{9 - 16}{4} \]
\[ = \frac{-7}{4} \]

### Step 4: Calculate the product of the squares of the roots
To find \( \alpha^2 \beta^2 \), we can use the identity:
\[ \alpha^2 \beta^2 = (\alpha \beta)^2 \]

Substituting the value of \( \alpha \beta \):
\[ \alpha^2 \beta^2 = (2)^2 = 4 \]

### Step 5: Form the new quadratic equation
The new 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:
\[ x^2 - \left(-\frac{7}{4}\right)x + 4 = 0 \]
\[ x^2 + \frac{7}{4}x + 4 = 0 \]

### Step 6: Clear the fractions
To eliminate the fraction, multiply the entire equation by 4:
\[ 4x^2 + 7x + 16 = 0 \]

### Final Answer
Thus, the equation whose roots are \( \alpha^2 \) and \( \beta^2 \) is:
\[ 4x^2 + 7x + 16 = 0 \]

### Conclusion
The correct option is:
**a. \( 4x^2 + 7x + 16 = 0 \)**