Solve for t.
23t2+45t–2=0
Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.


We are given the quadratic equation:

\[
23t^2 + 45t - 2 = 0
\]

To solve this quadratic equation, we will use the **quadratic formula**:

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

Here, the coefficients are:
- \(a = 23\)
- \(b = 45\)
- \(c = -2\)

### Step 1: Calculate the discriminant
The discriminant is the expression inside the square root:

\[
\Delta = b^2 - 4ac
\]

Substitute the values of \(a\), \(b\), and \(c\):

\[
\Delta = 45^2 - 4(23)(-2) = 2025 + 184 = 2209
\]

### Step 2: Solve for \(t\)
Now substitute the discriminant and the coefficients into the quadratic formula:

\[
t = \frac{-45 \pm \sqrt{2209}}{2(23)}
\]

Since \(\sqrt{2209} = 47\), we have:

\[
t = \frac{-45 \pm 47}{46}
\]

### Step 3: Calculate the two possible values for \(t\)

1. For the positive root:
\[
t_1 = \frac{-45 + 47}{46} = \frac{2}{46} = \frac{1}{23}
\]

2. For the negative root:
\[
t_2 = \frac{-45 - 47}{46} = \frac{-92}{46} = -2
\]

### Final Answer:
The two solutions are:

\[
t = \frac{1}{23}, -2
\]

Let me know if you'd like further clarification or more details!

Here are five related questions to further explore:
1. How can we verify that the solutions to a quadratic equation are correct?
2. What is the significance of the discriminant in a quadratic equation?
3. How do we know when a quadratic equation has no real solutions?
4. Can the quadratic formula be used for any quadratic equation, regardless of its coefficients?
5. How do we factor quadratics when factoring is simpler than using the quadratic formula?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) to determine if real, repeated, or complex solutions exist before proceeding with solving.

Solve for t. 23t^2 + 45t - 2 = 0. Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.

Learn how to solve the quadratic equation 23t^2 + 45t - 2 using the quadratic formula. This detailed guide walks through calculating the discriminant and finding the solutions in simplest form. Ideal for students in Grades 9-10 studying algebra and quadratic equations.

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