To find the roots of the equation [tex]\( x^2 + 2x - 5 = 0 \)[/tex], we need to solve it step-by-step. We will use the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
In this quadratic equation, the coefficients are:
[tex]\[
a = 1, \quad b = 2, \quad c = -5
\][/tex]
First, we calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
[tex]\[
\Delta = 2^2 - 4 \cdot 1 \cdot (-5)
\][/tex]
[tex]\[
\Delta = 4 + 20
\][/tex]
[tex]\[
\Delta = 24
\][/tex]
Now, we will use the discriminant to find the roots using the quadratic formula:
[tex]\[
x = \frac{{-2 \pm \sqrt{24}}}{2 \cdot 1}
\][/tex]
We know that [tex]\(\sqrt{24} = 2\sqrt{6}\)[/tex]. Therefore, substituting it back into the equation gives:
[tex]\[
x = \frac{{-2 \pm 2\sqrt{6}}}{2}
\][/tex]
We can simplify the expression:
[tex]\[
x = \frac{{-2}}{2} \pm \frac{{2\sqrt{6}}}{2}
\][/tex]
[tex]\[
x = -1 \pm \sqrt{6}
\][/tex]
Thus, the roots of the equation [tex]\( x^2 + 2x - 5 = 0 \)[/tex] are:
[tex]\[
x = -1 + \sqrt{6} \quad \text{and} \quad x = -1 - \sqrt{6}
\][/tex]
So, the correct answers are:
C. [tex]\(x = -1 + \sqrt{6}\)[/tex]
B. [tex]\(x = -1 - \sqrt{6}\)[/tex]
