Solve the equation using the Quadratic Formula.

Question

x^2 -7x-10
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1)x=____ (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Use a comma to separate answers as needed.)
2) There are no real solutions.

Answer 1

To solve the equation \(x^2 - 7x - 10 = 0\) using the Quadratic Formula, we start by identifying the coefficients:

• \(a = 1\)
• \(b = -7\)
• \(c = -10\)

The Quadratic Formula is given by:

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

Next, we calculate \(b^2 - 4ac\):

\[ b^2 - 4ac = (-7)^2 - 4(1)(-10) = 49 + 40 = 89 \]

Now we can substitute the values into the Quadratic Formula:

\[ x = \frac{-(-7) \pm \sqrt{89}}{2(1)} = \frac{7 \pm \sqrt{89}}{2} \]

This gives us two possible solutions:

\[ x = \frac{7 + \sqrt{89}}{2}, \quad x = \frac{7 - \sqrt{89}}{2} \]

Thus, the complete solution is:

1. \(x = \frac{7 + \sqrt{89}}{2}, \frac{7 - \sqrt{89}}{2}\)

So the final answer is:

\[ \text{1) } x = \frac{7 + \sqrt{89}}{2}, \frac{7 - \sqrt{89}}{2} \]

There are real solutions, and option 2) is not correct.