Find the equation whose roots are `(1)/(4)` times of the roots of the equation `x^(2)-3x+2=0`

Answer

To find the equation whose roots are \(\frac{1}{4}\) times the roots of the equation \(x^2 - 3x + 2 = 0\), we will follow these steps:

### Step 1: Find the roots of the given equation
The given quadratic equation is:
\[ x^2 - 3x + 2 = 0 \]
We can factor this equation:
\[ (x - 1)(x - 2) = 0 \]
Thus, the roots are:
\[ \alpha = 1 \quad \text{and} \quad \beta = 2 \]

### Step 2: Calculate the sum and product of the roots
Using Vieta's formulas:
- The sum of the roots \(\alpha + \beta\) is:
\[ \alpha + \beta = 1 + 2 = 3 \]
- The product of the roots \(\alpha \beta\) is:
\[ \alpha \beta = 1 \times 2 = 2 \]

### Step 3: Determine the new roots
The new roots are \(\frac{1}{4}\) times the original roots:
\[ \text{New roots} = \left(\frac{\alpha}{4}, \frac{\beta}{4}\right) = \left(\frac{1}{4}, \frac{2}{4}\right) = \left(\frac{1}{4}, \frac{1}{2}\right) \]

### Step 4: Calculate the sum and product of the new roots
- The sum of the new roots is:
\[ \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \]
- The product of the new roots is:
\[ \frac{1}{4} \times \frac{1}{2} = \frac{1}{8} \]

### Step 5: Form the new quadratic equation
Using the sum and product of the new roots, we can write the new quadratic equation in the standard form:
\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]
Substituting the values we found:
\[ x^2 - \frac{3}{4}x + \frac{1}{8} = 0 \]

### Step 6: Eliminate fractions by multiplying through by 8
To eliminate the fractions, multiply the entire equation by 8:
\[ 8x^2 - 6x + 1 = 0 \]

### Final Answer
The equation whose roots are \(\frac{1}{4}\) times the roots of the equation \(x^2 - 3x + 2 = 0\) is:
\[ 8x^2 - 6x + 1 = 0 \]

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