To solve for [tex]\( x \)[/tex] in the quadratic equation [tex]\( 2x^2 - 1 = 3x + 4 \)[/tex], follow these steps:
1. Rewrite the equation in standard form: A quadratic equation must be in the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
[tex]\[
2x^2 - 1 - 3x - 4 = 0
\][/tex]
Simplify to get:
[tex]\[
2x^2 - 3x - 5 = 0
\][/tex]
2. Identify coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a = 2, \quad b = -3, \quad c = -5
\][/tex]
3. Apply the quadratic formula: The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2}
\][/tex]
So, the correctly applied quadratic formula for this equation is:
[tex]\[
\boxed{x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)}}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
