Certainly! Let's construct the quadratic equations step-by-step for each pair of roots.
### 1. Roots: [tex]\(-3\)[/tex] and [tex]\(5\)[/tex]
A quadratic equation can be constructed from its roots using the form:
[tex]\[ (x - \text{root1})(x - \text{root2}) = 0 \][/tex]
For the roots [tex]\(-3\)[/tex] and [tex]\(5\)[/tex]:
[tex]\[ (x - (-3))(x - 5) = 0 \][/tex]
[tex]\[ (x + 3)(x - 5) = 0 \][/tex]
Now, expand the product:
[tex]\[ x^2 - 5x + 3x - 15 = 0 \][/tex]
[tex]\[ x^2 - 2x - 15 = 0 \][/tex]
So, the quadratic equation is:
[tex]\[ 1x^2 + (-2)x + (-15) = 0 \][/tex]
### 2. Roots: [tex]\(\frac{-6}{7}\)[/tex] and [tex]\(\frac{-2}{9}\)[/tex]
For the roots [tex]\(\frac{-6}{7}\)[/tex] and [tex]\(\frac{-2}{9}\)[/tex]:
[tex]\[ \left(x - \frac{-6}{7}\right)\left(x - \frac{-2}{9}\right) = 0 \][/tex]
[tex]\[ \left(x + \frac{6}{7}\right)\left(x + \frac{2}{9}\right) = 0 \][/tex]
Now, expand the product:
[tex]\[ x^2 + \left(\frac{6}{7} + \frac{2}{9}\right)x + \left(\frac{6}{7} \cdot \frac{2}{9}\right) = 0 \][/tex]
Calculate the coefficients:
[tex]\[ \frac{6}{7} + \frac{2}{9} = \frac{54}{63} + \frac{14}{63} = \frac{68}{63} \approx 1.079365079 \][/tex]
[tex]\[ \frac{6}{7} \cdot \frac{2}{9} = \frac{12}{63} = \frac{4}{21} \approx 0.190476190 \][/tex]
So, the quadratic equation is:
[tex]\[ 1x^2 + (1.0793650793650793)x + (0.19047619047619047) = 0 \][/tex]
### 3. Roots: [tex]\(-7\)[/tex] and [tex]\(13\)[/tex]
For the roots [tex]\(-7\)[/tex] and [tex]\(13\)[/tex]:
[tex]\[ (x - (-7))(x - 13) = 0 \][/tex]
[tex]\[ (x + 7)(x - 13) = 0 \][/tex]
Now, expand the product:
[tex]\[ x^2 - 13x + 7x - 91 = 0 \][/tex]
[tex]\[ x^2 - 6x - 91 = 0 \][/tex]
So, the quadratic equation is:
[tex]\[ 1x^2 + (-6)x + (-91) = 0 \][/tex]
### Summary
The quadratic equations for the given roots are:
1. [tex]\(1x^2 + (-2)x + (-15) = 0\)[/tex]
2. [tex]\(1x^2 + (1.0793650793650793)x + (0.19047619047619047) = 0\)[/tex]
3. [tex]\(1x^2 + (-6)x + (-91) = 0\)[/tex]
