Consider the quadratic equation `(log_10 8)x^2-(log_10 5)x=2(log_2 10)^-1 -x.` Which of the following quantities are irrational.

Options

A. sum of roots
B. product of roots
C. sum of coefficients
D. discriminant

Answer

To solve the given quadratic equation \( ( \log_{10} 8 ) x^2 - ( \log_{10} 5 ) x = 2 ( \log_{2} 10 )^{-1} - x \), we will first rearrange the equation into standard quadratic form and then analyze the properties of the coefficients and roots.

### Step 1: Rearranging the Equation
The given equation can be rearranged as follows:

\[ ( \log_{10} 8 ) x^2 - ( \log_{10} 5 + 1 ) x + 2 ( \log_{2} 10 )^{-1} = 0 \]

### Step 2: Identifying Coefficients
From the standard form \( ax^2 + bx + c = 0 \), we identify:
- \( a = \log_{10} 8 \)
- \( b = -(\log_{10} 5 + 1) \)
- \( c = 2 ( \log_{2} 10 )^{-1} \)

### Step 3: Sum of Roots
The sum of the roots of a quadratic equation is given by the formula:

\[ \text{Sum of roots} = -\frac{b}{a} \]

Substituting the values of \( b \) and \( a \):

\[ \text{Sum of roots} = -\frac{-(\log_{10} 5 + 1)}{\log_{10} 8} = \frac{\log_{10} 5 + 1}{\log_{10} 8} \]

Using the property of logarithms, we can express \( \log_{10} 8 \) as \( 3 \log_{10} 2 \):

\[ \text{Sum of roots} = \frac{\log_{10} 5 + 1}{3 \log_{10} 2} \]

### Step 4: Product of Roots
The product of the roots is given by:

\[ \text{Product of roots} = \frac{c}{a} \]

Substituting the values of \( c \) and \( a \):

\[ \text{Product of roots} = \frac{2 ( \log_{2} 10 )^{-1}}{\log_{10} 8} \]

Using the change of base formula, we can express \( \log_{2} 10 \) as \( \frac{1}{\log_{10} 2} \):

\[ \text{Product of roots} = \frac{2 \log_{10} 2}{\log_{10} 8} = \frac{2 \log_{10} 2}{3 \log_{10} 2} = \frac{2}{3} \]

### Step 5: Discriminant
The discriminant \( D \) of the quadratic equation is given by:

\[ D = b^2 - 4ac \]

Calculating \( D \):

\[ D = (-(\log_{10} 5 + 1))^2 - 4 \cdot \log_{10} 8 \cdot 2 ( \log_{2} 10 )^{-1} \]

Substituting values:

\[ D = (\log_{10} 5 + 1)^2 - 8 \cdot ( \log_{2} 10 )^{-1} \cdot \log_{10} 8 \]

### Step 6: Sum of Coefficients
The sum of the coefficients \( a + b + c \):

\[ \text{Sum of coefficients} = \log_{10} 8 - (\log_{10} 5 + 1) + 2 ( \log_{2} 10 )^{-1} \]

### Conclusion
Now we need to determine which of these quantities are irrational:

1. **Sum of Roots**: \( \frac{\log_{10} 5 + 1}{3 \log_{10} 2} \) - This is rational since it is a ratio of logarithms.
2. **Product of Roots**: \( \frac{2}{3} \) - This is rational.
3. **Discriminant**: This expression is complex and likely irrational.
4. **Sum of Coefficients**: This expression is also complex and likely irrational.

### Final Answer
The quantities that are irrational are:
- Discriminant
- Sum of Coefficients