Question

which of the following quadratic equations has roots of 2 + i and 2 - i? equation 1: (x^{2}-4x + 5=0) equation 2: (x^{2}+4x + 5=0) equation 3: (x^{2}-4x - 5=0) (1 point) equation □

Explanation:

Step1: Recall quadratic - equation formula

For a quadratic equation \(ax^{2}+bx + c = 0\) with roots \(r_1\) and \(r_2\), the sum of the roots \(r_1 + r_2=-\frac{b}{a}\) and the product of the roots \(r_1r_2=\frac{c}{a}\).

Step2: Calculate sum and product of given roots

Given \(r_1 = 2 + i\) and \(r_2=2 - i\).
The sum \(r_1 + r_2=(2 + i)+(2 - i)=4\).
The product \(r_1r_2=(2 + i)(2 - i)=2^{2}-i^{2}=4+1 = 5\).

Step3: Write the quadratic equation

A quadratic equation with roots \(r_1\) and \(r_2\) is \(x^{2}-(r_1 + r_2)x+r_1r_2 = 0\).
Substituting \(r_1 + r_2 = 4\) and \(r_1r_2 = 5\), we get \(x^{2}-4x + 5=0\).

Answer:

Equation 1