Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( t = -\frac{1}{2} \) or \( t = -\frac{2}{3} \).
Solution
We start with the equation
\[
6t^2 + 7t + 2 = 0.
\]
**Step 1: Identify the coefficients**
The quadratic equation has the standard form
\[
at^2 + bt + c = 0,
\]
with
\[
a = 6, \quad b = 7, \quad c = 2.
\]
**Step 2: Compute the discriminant**
The discriminant is given by
\[
\Delta = b^2 - 4ac.
\]
Substitute the values:
\[
\Delta = 7^2 - 4 \cdot 6 \cdot 2 = 49 - 48 = 1.
\]
**Step 3: Apply the quadratic formula**
The quadratic formula is
\[
t = \frac{-b \pm \sqrt{\Delta}}{2a}.
\]
Substitute the values:
\[
t = \frac{-7 \pm \sqrt{1}}{2 \cdot 6} = \frac{-7 \pm 1}{12}.
\]
**Step 4: Find the two solutions**
1. For the positive square root:
\[
t = \frac{-7 + 1}{12} = \frac{-6}{12} = -\frac{1}{2}.
\]
2. For the negative square root:
\[
t = \frac{-7 - 1}{12} = \frac{-8}{12} = -\frac{2}{3}.
\]
**Final Answer:**
\[
t = -\frac{1}{2} \quad \text{or} \quad t = -\frac{2}{3}.
\]
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Simplify this solution Bonus Knowledge
To solve the equation \( 6t^{2} + 7t + 2 = 0 \), we can use the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 6 \), \( b = 7 \), and \( c = 2 \). **Step 1**: Calculate the discriminant \( b^2 - 4ac \). \[ b^2 - 4ac = 7^2 - 4 \cdot 6 \cdot 2 = 49 - 48 = 1 \] **Step 2**: Use the quadratic formula. \[ t = \frac{-7 \pm \sqrt{1}}{2 \cdot 6} = \frac{-7 \pm 1}{12} \] **Step 3**: Calculate the two possible values for \( t \). 1. \( t = \frac{-7 + 1}{12} = \frac{-6}{12} = -\frac{1}{2} \) 2. \( t = \frac{-7 - 1}{12} = \frac{-8}{12} = -\frac{2}{3} \) Thus, the solutions for \( t \) are \[ t = -\frac{1}{2} \quad \text{and} \quad t = -\frac{2}{3} \]
