Question

What are the exact solutions of x2 − x − 4 = 0, where x equals negative b plus or minus the square root of b squared minus 4 times a times c all over 2 times a?

x = the quantity of negative 1 plus or minus the square root of 15 all over 2
x = the quantity of 1 plus or minus the square root of 15 all over 2
x = the quantity of 1 plus or minus the square root of 17 all over 2
x = the quantity of negative 1 plus or minus the square root of 17 all over 2

Answer 1

To solve the quadratic equation \(x^2 - x - 4 = 0\) using the quadratic formula, we identify the coefficients \(a\), \(b\), and \(c\):

- \(a = 1\)
- \(b = -1\)
- \(c = -4\)

The quadratic formula is given by:

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

We first compute \(b^2 - 4ac\):

\[
b^2 = (-1)^2 = 1
\]
\[
4ac = 4 \cdot 1 \cdot (-4) = -16
\]
\[
b^2 - 4ac = 1 - (-16) = 1 + 16 = 17
\]

Now we can substitute \(b\), \(b^2 - 4ac\), and \(a\) back into the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{17}}{2 \cdot 1} = \frac{1 \pm \sqrt{17}}{2}
\]

So the exact solutions to the equation \(x^2 - x - 4 = 0\) are:

\[
x = \frac{1 + \sqrt{17}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{17}}{2}
\]

Now, looking at the provided options, the correct answer is:

**\(x = \frac{1 \pm \sqrt{17}}{2}\)**.