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Question 2:

A quadratic equation whose roots are [tex]\(2 + \sqrt{5}\)[/tex] and [tex]\(2 - \sqrt{5}\)[/tex] is:

A. [tex]\(x^2 - 4x - 1 = 0\)[/tex]

B. [tex]\(x^2 + 4x - 1 = 0\)[/tex]

C. [tex]\(x^2 - 4x + 1 = 0\)[/tex]

D. [tex]\(x^2 + 4x + 1 = 0\)[/tex]



Answer :

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]