To find the quadratic equation whose roots are [tex]\( 5 \)[/tex] and [tex]\( \frac{-1}{3} \)[/tex], we will follow these steps:
1. Understand the Form of the Quadratic Equation:
A quadratic equation with roots [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex] can be written in the form:
[tex]\[
x^2 - (\alpha + \beta)x + \alpha\beta = 0
\][/tex]
Here, [tex]\( \alpha = 5 \)[/tex] and [tex]\( \beta = \frac{-1}{3} \)[/tex].
2. Calculate the Sum of the Roots:
[tex]\[
\alpha + \beta = 5 + \left(\frac{-1}{3}\right) = 5 - \frac{1}{3} = \frac{15}{3} - \frac{1}{3} = \frac{14}{3}
\][/tex]
3. Calculate the Product of the Roots:
[tex]\[
\alpha \cdot \beta = 5 \cdot \left(\frac{-1}{3}\right) = \frac{-5}{3}
\][/tex]
4. Form the Quadratic Equation:
Substitute the sum and product of the roots into the standard quadratic equation form:
[tex]\[
x^2 - \left(\frac{14}{3}\right)x + \left(\frac{-5}{3}\right) = 0
\][/tex]
5. Simplify the Equation if Necessary:
Sometimes it is preferable to clear the fractions by multiplying through by 3 (the denominator):
[tex]\[
3 \left( x^2 - \left(\frac{14}{3}\right)x + \left(\frac{-5}{3}\right) \right) = 0
\][/tex]
This results in:
[tex]\[
3x^2 - 14x - 5 = 0
\][/tex]
Therefore, the quadratic equation whose roots are 5 and [tex]\(\frac{-1}{3}\)[/tex] is:
[tex]\[ \boxed{3x^2 - 14x - 5 = 0} \][/tex]
