To solve the quadratic equation [tex]\(4x^2 - x - 5 = 0\)[/tex] using the quadratic formula, we use the formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 4\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -5\)[/tex].
First, we compute the discriminant, which is given by:
[tex]\[
b^2 - 4ac
\][/tex]
Substituting [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
(-1)^2 - 4 \cdot 4 \cdot (-5) = 1 + 80 = 81
\][/tex]
So, the discriminant is 81.
Next, we apply the quadratic formula to find the solutions:
[tex]\[
x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 4} = \frac{1 \pm 9}{8}
\][/tex]
This yields two possible solutions:
[tex]\[
x_1 = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4} = 1.25
\][/tex]
[tex]\[
x_2 = \frac{1 - 9}{8} = \frac{-8}{8} = -1
\][/tex]
Thus, the solutions to the quadratic equation [tex]\(4x^2 - x - 5 = 0\)[/tex] are [tex]\(x_1 = 1.25\)[/tex] and [tex]\(x_2 = -1\)[/tex].
Next, let's check the given options to see which of them apply:
A. [tex]\(\frac{5}{4}\)[/tex] [tex]\(\Rightarrow\)[/tex] Correct, as [tex]\(\frac{5}{4} = 1.25\)[/tex]
B. [tex]\(\frac{3}{2}\)[/tex] [tex]\(\Rightarrow\)[/tex] Incorrect
C. [tex]\(1\)[/tex] [tex]\(\Rightarrow\)[/tex] Incorrect
D. [tex]\(-1\)[/tex] [tex]\(\Rightarrow\)[/tex] Correct
E. [tex]\(\frac{-4}{5}\)[/tex] [tex]\(\Rightarrow\)[/tex] Incorrect
F. [tex]\(\frac{2}{3}\)[/tex] [tex]\(\Rightarrow\)[/tex] Incorrect
Therefore, the correct answers are:
- [tex]\( \boxed{\frac{5}{4}} \)[/tex]
- [tex]\( \boxed{-1} \)[/tex]
