To solve the quadratic equation \( 15x^2 - 23x - 28 = 0 \), we can use the quadratic formula:

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

where \( a = 15 \), \( b = -23 \), and \( c = -28 \).

### Step 1: Calculate the Discriminant

\[
\text{Discriminant} = b^2 - 4ac
\]

\[
\text{Discriminant} = (-23)^2 - 4 \cdot 15 \cdot (-28)
\]

\[
\text{Discriminant} = 529 + 1680
\]

\[
\text{Discriminant} = 2209
\]

### Step 2: Solve for \( x \) using the Quadratic Formula

Since the discriminant is positive, there are two real and distinct solutions.

\[
x = \frac{{-(-23) \pm \sqrt{{2209}}}}{2 \cdot 15}
\]

\[
x = \frac{{23 \pm 47}}{30}
\]

### Step 3: Calculate the Two Solutions

**Solution 1:**

\[
x = \frac{{23 + 47}}{30}
\]

\[
x = \frac{70}{30}
\]

\[
x = \frac{7}{3}
\]

**Solution 2:**

\[
x = \frac{{23 - 47}}{30}
\]

\[
x = \frac{-24}{30}
\]

\[
x = -\frac{4}{5}
\]

### Final Solutions

The solutions to the quadratic equation \( 15x^2 - 23x - 28 = 0 \) are:

\[
x = \frac{7}{3} \quad \text{and} \quad x = -\frac{4}{5}
\]

---

**Would you like to know more details or do you have any other questions? Here are some additional problems related to quadratic equations:**

1. **Solve** \(2x^2 - 5x + 2 = 0\).
2. **Find the vertex** of the parabola \(y = 3x^2 - 12x + 7\).
3. **Determine the axis of symmetry** for the function \(f(x) = 4x^2 - 8x + 5\).
4. **Solve by factoring** \(x^2 - 9x + 20 = 0\).
5. **Identify the nature of the roots** for \(5x^2 + 3x + 7 = 0\).
6. **Complete the square** to solve \(x^2 + 6x + 5 = 0\).
7. **Graph the equation** \(y = -2x^2 + 8x - 6\) and find its intercepts.
8. **Find the roots** of the equation \(7x^2 - 14x = 0\) using the quadratic formula.

**Tip:** When solving quadratic equations, always check if it can be factored easily before using the quadratic formula.

Learn how to solve the quadratic equation 15x^2 - 23x - 28 step-by-step using the quadratic formula. This problem involves calculating the discriminant and finding two real solutions. Suitable for students in Grades 10-12.

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