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Use the quadratic formula to solve. Express your answer in simplest form.
13, t, squared, minus, 8, t, plus, 1, equals, minus, 3, t, squared
13t
2
−8t+1=
−3t
2

Answer
Attempt 1 out of 5

Additional Solution
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Answer 1

To solve the equation \(13t^2 - 8t + 1 = -3t^2\), we first need to rearrange the equation to set it to zero.

1. Add \(3t^2\) to both sides:
\[
13t^2 + 3t^2 - 8t + 1 = 0
\]
This simplifies to:
\[
16t^2 - 8t + 1 = 0
\]

Now, we have a standard quadratic equation in the form \(at^2 + bt + c = 0\), where:
- \(a = 16\)
- \(b = -8\)
- \(c = 1\)

Next, we will use the quadratic formula, which is given by:
\[
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

2. Calculate the discriminant \(b^2 - 4ac\):
\[
b^2 = (-8)^2 = 64
\]
\[
4ac = 4 \cdot 16 \cdot 1 = 64
\]
Therefore, the discriminant is:
\[
64 - 64 = 0
\]

Since the discriminant is \(0\), there is exactly one solution.

3. Substitute into the quadratic formula:
\[
t = \frac{-(-8) \pm \sqrt{0}}{2 \cdot 16}
\]
This simplifies to:
\[
t = \frac{8}{32} = \frac{1}{4}
\]

Thus, the solution to the equation \(13t^2 - 8t + 1 = -3t^2\) is:
\[
\boxed{\frac{1}{4}}
\]