To solve the equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex], we can use the quadratic formula which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -3 \)[/tex].
1. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
[tex]\[
= 3^2 - 4 \times 1 \times (-3)
\][/tex]
[tex]\[
= 9 + 12
\][/tex]
[tex]\[
= 21
\][/tex]
2. Find the roots using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a}
\][/tex]
[tex]\[
= \frac{-3 \pm \sqrt{21}}{2 \times 1}
\][/tex]
[tex]\[
= \frac{-3 \pm \sqrt{21}}{2}
\][/tex]
Thus, the two roots of the equation [tex]\( x^2 + 3x - 3 = 0 \)[/tex] are:
[tex]\[
x = \frac{-3 + \sqrt{21}}{2}
\][/tex]
[tex]\[
x = \frac{-3 - \sqrt{21}}{2}
\][/tex]
Therefore, the correct answers are:
D. [tex]\( x = \frac{-3 + \sqrt{21}}{2} \)[/tex]
B. [tex]\( x = \frac{-3 - \sqrt{21}}{2} \)[/tex]
