Find the equation whose roots are reciprocals of the roots of `5x^(2)+6x+7=0`.

Answer

To find the equation whose roots are the reciprocals of the roots of the quadratic equation \(5x^2 + 6x + 7 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here, we have:
- \(a = 5\)
- \(b = 6\)
- \(c = 7\)

### Step 2: Calculate the sum and product of the roots
Let the roots of the equation be \(\alpha\) and \(\beta\).
- The sum of the roots \(\alpha + \beta\) is given by the formula:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{6}{5} \]
- The product of the roots \(\alpha \beta\) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{7}{5} \]

### Step 3: Find the sum and product of the reciprocals of the roots
The roots we are looking for are \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\).
- The sum of the reciprocals of the roots is:
\[ \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta} = \frac{-\frac{6}{5}}{\frac{7}{5}} = -\frac{6}{7} \]
- The product of the reciprocals of the roots is:
\[ \frac{1}{\alpha} \cdot \frac{1}{\beta} = \frac{1}{\alpha \beta} = \frac{1}{\frac{7}{5}} = \frac{5}{7} \]

### Step 4: Form the new quadratic equation
Using the sum and product of the new roots, we can form the quadratic equation:
\[ x^2 - \left(\text{sum of roots}\right)x + \left(\text{product of roots}\right) = 0 \]
Substituting the values we found:
\[ x^2 - \left(-\frac{6}{7}\right)x + \frac{5}{7} = 0 \]
This simplifies to:
\[ x^2 + \frac{6}{7}x + \frac{5}{7} = 0 \]

### Step 5: Eliminate the fraction by multiplying through by 7
To eliminate the fractions, multiply the entire equation by 7:
\[ 7x^2 + 6x + 5 = 0 \]

### Final Answer
The equation whose roots are the reciprocals of the roots of \(5x^2 + 6x + 7 = 0\) is:
\[ \boxed{7x^2 + 6x + 5 = 0} \]

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