First, let's rearrange the given quadratic equation into the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. We start with:
[tex]\[
4x^2 - 5 = 3x + 4
\][/tex]
To convert this into standard form, move all terms to one side of the equation:
[tex]\[
4x^2 - 5 - 3x - 4 = 0
\][/tex]
Combine like terms:
[tex]\[
4x^2 - 3x - 9 = 0
\][/tex]
Now it's in standard form, where [tex]\(a = 4\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -9\)[/tex].
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
1. Identify the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] again:
- [tex]\(a = 4\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = -9\)[/tex]
2. Substitute these values into the quadratic formula:
- The term [tex]\(-b\)[/tex] becomes [tex]\(-(-3) = 3\)[/tex].
- The term [tex]\(b^2\)[/tex] is [tex]\((-3)^2 = 9\)[/tex].
- The term [tex]\(4ac\)[/tex] is [tex]\(4 \cdot 4 \cdot -9 = -144\)[/tex].
Putting it all together, the discriminant [tex]\( \Delta \)[/tex] is:
[tex]\[
\Delta = b^2 - 4ac = 9 - (-144) = 9 + 144 = 153
\][/tex]
3. So, the quadratic formula setup becomes:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4}
\][/tex]
The correct setup is:
[tex]\[
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4}
\][/tex]
Therefore, the correct answer is:
D. [tex]\( x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(4)(-9)}}{2(4)} \)[/tex]
