To determine the quadratic equation whose roots are [tex]\(2 + \sqrt{5}\)[/tex] and [tex]\(2 - \sqrt{5}\)[/tex], let's follow these steps:
1. Identify the Roots:
The given roots are [tex]\( \alpha = 2 + \sqrt{5} \)[/tex] and [tex]\( \beta = 2 - \sqrt{5} \)[/tex].
2. Sum of the Roots:
The sum of the roots for a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\( -\frac{b}{a} \)[/tex].
[tex]\[
\alpha + \beta = (2 + \sqrt{5}) + (2 - \sqrt{5}) = 2 + 2 = 4
\][/tex]
3. Product of the Roots:
The product of the roots for a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by [tex]\( \frac{c}{a} \)[/tex].
[tex]\[
\alpha \beta = (2 + \sqrt{5})(2 - \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1
\][/tex]
4. Form the Quadratic Equation:
Using Vieta's formulas, the quadratic equation with given roots [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] can be written as:
[tex]\[
x^2 - (sum\ of\ roots)x + (product\ of\ roots) = 0
\][/tex]
Substituting the sum and product of the roots into the equation:
[tex]\[
x^2 - 4x - 1 = 0
\][/tex]
Therefore, the quadratic equation with roots [tex]\(2+\sqrt{5}\)[/tex] and [tex]\(2-\sqrt{5}\)[/tex] is:
[tex]\[
x^2 - 4x - 1 = 0
\][/tex]
Thus, the correct answer is:
A. [tex]\(x^2 - 4 x - 1 = 0\)[/tex]
