Given that the sum of the zeroes of a quadratic polynomial is [tex]\(\frac{4}{5}\)[/tex] and their product is [tex]\(-\frac{1}{2}\)[/tex], we need to find the quadratic polynomial.
Let's denote the quadratic polynomial as [tex]\(ax^2 + bx + c\)[/tex]. According to the properties of quadratic polynomials, we know the following:
1. The sum of the roots (zeroes) is given by [tex]\(-\frac{b}{a}\)[/tex].
2. The product of the roots (zeroes) is given by [tex]\(\frac{c}{a}\)[/tex].
Given:
- Sum of zeroes = [tex]\(\frac{4}{5}\)[/tex]
- Product of zeroes = [tex]\(-\frac{1}{2}\)[/tex]
We can use these relationships to find the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] of the quadratic polynomial. For simplicity, we can let [tex]\(a = 1\)[/tex]. This simplifies our polynomial to the form [tex]\(x^2 + bx + c\)[/tex].
Using the given sum of the zeroes:
[tex]\[
-\frac{b}{1} = \frac{4}{5}
\][/tex]
[tex]\[
b = -\frac{4}{5}
\][/tex]
Using the given product of the zeroes:
[tex]\[
\frac{c}{1} = -\frac{1}{2}
\][/tex]
[tex]\[
c = -\frac{1}{2}
\][/tex]
Therefore, with [tex]\(a = 1\)[/tex], [tex]\(b = -\frac{4}{5}\)[/tex], and [tex]\(c = -\frac{1}{2}\)[/tex], the quadratic polynomial is:
[tex]\[
x^2 - \frac{4}{5}x - \frac{1}{2}
\][/tex]
Thus, the quadratic polynomial with the given sum and product of zeroes is:
[tex]\[
x^2 - 0.8x - 0.5
\][/tex]
