Let's determine which of the provided options correctly represents the quadratic formula. The quadratic formula is used to find the solutions to a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
The standard quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's compare this standard formula to each of the given options:
1. [tex]\( x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \)[/tex]
This option only includes the positive square root, which would only provide one of the two possible solutions. Therefore, this is not fully correct.
2. [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]
This option includes both the positive and negative square roots, represented by the [tex]\(\pm\)[/tex] symbol, which allows for both solutions to be found. This matches the standard quadratic formula.
3. [tex]\( x = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \)[/tex]
This option only includes the negative square root, which would again only provide one of the two possible solutions. Therefore, this is not fully correct.
4. [tex]\( x = \frac{b - \sqrt{b^2 - 4ac}}{2a} \)[/tex]
This option is incorrect as it begins with [tex]\( b \)[/tex] instead of [tex]\( -b \)[/tex], thus not matching the standard quadratic formula.
Given these comparisons, the correct representation of the quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
