Question

Solve the equation for all real solutions in simplest form.
k, squared, plus, 5, k, minus, 3, equals, 0
k
2
+5k−3=
0

Answer 1

To solve the equation \( k^2 + 5k - 3 = 0 \), we can use the quadratic formula:

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

where \( a = 1 \), \( b = 5 \), and \( c = -3 \).

First, we calculate the discriminant:

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

Now, substituting into the quadratic formula:

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

Thus, the solutions for \( k \) are:

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

These are the two real solutions in simplest form:

\[
k = \frac{-5 + \sqrt{37}}{2}, \quad k = \frac{-5 - \sqrt{37}}{2}
\]