Question

Solve by the quadratic formula and simplify.
4 x squared minus x minus 5 equals 04x2−x−5=0

Answer 1

To solve the quadratic equation \( 4x^2 - x - 5 = 0 \) using the quadratic formula, we can use the formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 4 \), \( b = -1 \), and \( c = -5 \).

First, we need to calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = (-1)^2 = 1
\]
\[
4ac = 4 \cdot 4 \cdot (-5) = -80
\]
\[
b^2 - 4ac = 1 - (-80) = 1 + 80 = 81
\]

Now we can use the quadratic formula to find the roots:

\[
x = \frac{-(-1) \pm \sqrt{81}}{2 \cdot 4}
\]
\[
x = \frac{1 \pm 9}{8}
\]

This gives us two possible solutions:

1. \( x = \frac{1 + 9}{8} = \frac{10}{8} = \frac{5}{4} \)
2. \( x = \frac{1 - 9}{8} = \frac{-8}{8} = -1 \)

Thus, the two solutions to the equation \( 4x^2 - x - 5 = 0 \) are:

\[
x = \frac{5}{4} \quad \text{and} \quad x = -1
\]