To determine the correct equation that applies the quadratic formula, we need to start by rearranging the given quadratic equation [tex]\(2x^2 - 1 = 3x + 4\)[/tex] into the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
Here are the steps:
1. Start with the original equation:
[tex]\[
2x^2 - 1 = 3x + 4
\][/tex]
2. Move all terms to one side to bring it to the standard form:
[tex]\[
2x^2 - 1 - 3x - 4 = 0
\][/tex]
3. Simplify the equation:
[tex]\[
2x^2 - 3x - 5 = 0
\][/tex]
Now we have the quadratic equation in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -5\)[/tex].
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the formula:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -5\)[/tex]
This gives us:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}
\][/tex]
Simplify the expression further:
- The numerator should be:
[tex]\[
-(-3) = 3
\][/tex]
- The discriminant inside the square root:
[tex]\[
(-3)^2 - 4 \cdot 2 \cdot (-5) = 9 + 40 = 49
\][/tex]
- The denominator:
[tex]\[
2 \cdot 2 = 4
\][/tex]
Thus, the correct equation applying the quadratic formula is:
[tex]\[
x = \frac{3 \pm \sqrt{49}}{4}
\][/tex]
Therefore, the answer choice that correctly applies the quadratic formula is:
D. [tex]\(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}\)[/tex]
