To construct a quadratic equation given its roots [tex]\(\frac{-6}{7}\)[/tex] and [tex]\(\frac{-2}{9}\)[/tex], we follow these steps:
1. Identify the roots:
- Root 1: [tex]\(\frac{-6}{7}\)[/tex]
- Root 2: [tex]\(\frac{-2}{9}\)[/tex]
2. Sum of the Roots:
The sum of the roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(-\frac{b}{a}\)[/tex].
The sum is therefore:
[tex]\[
\frac{-6}{7} + \frac{-2}{9}
\][/tex]
By finding a common denominator (63), we have:
[tex]\[
\frac{-6}{7} = \frac{-6 \times 9}{7 \times 9} = \frac{-54}{63}
\][/tex]
[tex]\[
\frac{-2}{9} = \frac{-2 \times 7}{9 \times 7} = \frac{-14}{63}
\][/tex]
Adding these fractions:
[tex]\[
\frac{-54}{63} + \frac{-14}{63} = \frac{-54 + (-14)}{63} = \frac{-68}{63} \approx -1.0793650793650793
\][/tex]
3. Product of the Roots:
The product of the roots of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\(\frac{c}{a}\)[/tex].
The product is therefore:
[tex]\[
\left(\frac{-6}{7}\right) \times \left(\frac{-2}{9}\right)
\][/tex]
Multiplying these fractions:
[tex]\[
\frac{(-6) \times (-2)}{7 \times 9} = \frac{12}{63} = \frac{4}{21} \approx 0.19047619047619047
\][/tex]
4. Construct the Quadratic Equation:
The standard form of the quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex].
Here, we know the following:
[tex]\[
a = 1
\][/tex]
[tex]\[
b = -(\text{sum of the roots}) = -\left(-1.0793650793650793\right) \approx 1.0793650793650793
\][/tex]
[tex]\[
c = \text{product of the roots} \approx 0.19047619047619047
\][/tex]
Substituting these into the standard quadratic form:
[tex]\[
x^2 + 1.0793650793650793x + 0.19047619047619047 = 0
\][/tex]
Therefore, the quadratic equation with roots [tex]\(\frac{-6}{7}\)[/tex] and [tex]\(\frac{-2}{9}\)[/tex] is:
[tex]\[
x^2 + 1.0793650793650793x + 0.19047619047619047 = 0
\][/tex]
