Use the quadratic formula to solve: 2$$ t^2 $$-4t-3=0

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To solve the quadratic equation $ 2t^2 - 4t - 3 = 0 $ using the quadratic formula, we use:

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

Here, $ a = 2 $, $ b = -4 $, and $ c = -3 $.

### Step 1: Calculate the Discriminant
First, calculate the discriminant ($ \Delta $):

\[
\Delta = b^2 - 4ac
\]
\[
\Delta = (-4)^2 - 4(2)(-3)
\]
\[
\Delta = 16 + 24 = 40
\]

### Step 2: Apply the Quadratic Formula
Now substitute the values into the quadratic formula:

\[
t = \frac{-(-4) \pm \sqrt{40}}{2(2)}
\]
\[
t = \frac{4 \pm \sqrt{40}}{4}
\]

### Step 3: Simplify the Square Root
Simplify $ \sqrt{40} $:

\[
\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}
\]

Substitute this back into the equation:

\[
t = \frac{4 \pm 2\sqrt{10}}{4}
\]

### Step 4: Simplify the Expression
Divide the numerator by the denominator:

\[
t = \frac{4}{4} \pm \frac{2\sqrt{10}}{4}
\]
\[
t = 1 \pm \frac{\sqrt{10}}{2}
\]

### Final Answers:
\[
t = 1 + \frac{\sqrt{10}}{2} \quad \text{and} \quad t = 1 - \frac{\sqrt{10}}{2}
\]

Would you like more details, or do you have any questions?

Here are 5 related questions you might find useful:
1. How do you simplify square roots when solving quadratic equations?
2. What is the discriminant, and how does it determine the nature of the roots?
3. Can the quadratic formula be used for all quadratic equations?
4. What are the different methods to solve quadratic equations?
5. How do you interpret the solutions of a quadratic equation on a graph?

**Tip:** Always simplify the square root and the fractions separately to avoid errors when solving quadratic equations using the formula.

Learn how to solve the quadratic equation 2t^2 - 4t - 3 using the quadratic formula with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and discriminant calculation. Suitable for students in Grades 9-12.

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