Question

Solve the equation for all real solutions in simplest form.
minus, 2, t, squared, plus, 7, t, minus, 4, equals, minus, 3, t, squared
−2t
2
+7t−4=
−3t
2

Answer 1

Let's start with the equation you provided:

\[
-2t^2 + 7t - 4 = -3t^2
\]

First, let's move all terms to one side of the equation. We can do this by adding \(3t^2\) to both sides:

\[
-2t^2 + 3t^2 + 7t - 4 = 0
\]

This simplifies to:

\[
t^2 + 7t - 4 = 0
\]

Now, we can solve this quadratic equation using the quadratic formula, which is given by:

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

In our equation, \(a = 1\), \(b = 7\), and \(c = -4\). Now, let's compute the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = 7^2 - 4(1)(-4) = 49 + 16 = 65
\]

Now plug the values into the quadratic formula:

\[
t = \frac{-7 \pm \sqrt{65}}{2(1)} = \frac{-7 \pm \sqrt{65}}{2}
\]

So, the solutions are:

\[
t = \frac{-7 + \sqrt{65}}{2} \quad \text{and} \quad t = \frac{-7 - \sqrt{65}}{2}
\]

Therefore, the real solutions to the equation are:

\[
t = \frac{-7 + \sqrt{65}}{2}, \quad t = \frac{-7 - \sqrt{65}}{2}
\]