Question

Solve the equation for all real solutions in simplest form.
t, squared, plus, 2, t, minus, 15, equals, minus, 3, t
t
2
+2t−15=
−3t

Answer 1

To solve the equation \( t^2 + 2t - 15 = -3t \), we first rearrange it by adding \( 3t \) to both sides:

\[
t^2 + 2t + 3t - 15 = 0
\]

This simplifies to:

\[
t^2 + 5t - 15 = 0
\]

Next, we can solve this quadratic equation using the quadratic formula, which is:

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

Here, \( a = 1 \), \( b = 5 \), and \( c = -15 \). Now, we calculate the discriminant:

\[
b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot (-15) = 25 + 60 = 85
\]

Now we can apply the quadratic formula:

\[
t = \frac{-5 \pm \sqrt{85}}{2 \cdot 1}
\]

This gives us:

\[
t = \frac{-5 \pm \sqrt{85}}{2}
\]

Thus, the two real solutions are:

\[
t = \frac{-5 + \sqrt{85}}{2} \quad \text{and} \quad t = \frac{-5 - \sqrt{85}}{2}
\]

These two solutions can be left in this form, which is their simplest form.