Solve the equation using the Quadratic Formula x^2 -5x -3=0

Question

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 \( x^2 - 5x - 3 = 0 \) using the Quadratic Formula, we can use the formula:

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

For the equation \( ax^2 + bx + c = 0 \), we have:

• \( a = 1 \)
• \( b = -5 \)
• \( c = -3 \)

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

\[ b^2 - 4ac = (-5)^2 - 4(1)(-3) = 25 + 12 = 37 \]

Since \( b^2 - 4ac \) is positive, we will have two real solutions. Now, substituting the values into the Quadratic Formula:

\[ x = \frac{-(-5) \pm \sqrt{37}}{2(1)} = \frac{5 \pm \sqrt{37}}{2} \]

Thus, the two solutions are:

\[ x = \frac{5 + \sqrt{37}}{2} \quad \text{and} \quad x = \frac{5 - \sqrt{37}}{2} \]

The final answer is:

1. \( x = \frac{5 + \sqrt{37}}{2}, \frac{5 - \sqrt{37}}{2} \)

2. There are no real solutions (this option is incorrect as we found real solutions).