To solve the quadratic equation \( 2x^2 + 3x - 9 = 0 \), we can use the **quadratic formula**:

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

In this equation, \( a = 2 \), \( b = 3 \), and \( c = -9 \).

### Step 1: Calculate the discriminant
The discriminant is the part under the square root: \( b^2 - 4ac \).

\[
\Delta = 3^2 - 4(2)(-9) = 9 + 72 = 81
\]

### Step 2: Apply the quadratic formula
Substitute \( a \), \( b \), and \( \Delta \) into the formula:

\[
x = \frac{-3 \pm \sqrt{81}}{2(2)} = \frac{-3 \pm 9}{4}
\]

### Step 3: Find the two solutions
1. For \( x_1 \):

\[
x_1 = \frac{-3 + 9}{4} = \frac{6}{4} = 1.5
\]

2. For \( x_2 \):

\[
x_2 = \frac{-3 - 9}{4} = \frac{-12}{4} = -3
\]

### Final solutions:
\[
x_1 = 1.5 \quad \text{and} \quad x_2 = -3
\]

Do you want more details or have any questions about this process?

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Here are 5 related questions for deeper practice:
1. Solve the equation \( 3x^2 - 5x + 2 = 0 \).
2. What is the discriminant of the equation \( 4x^2 + 12x + 9 = 0 \)?
3. Solve the quadratic equation \( 5x^2 - 3x - 8 = 0 \).
4. Can a quadratic equation have no real solutions? Under what conditions?
5. How does the value of the discriminant affect the number and nature of solutions?

**Tip:** The discriminant \( \Delta = b^2 - 4ac \) tells you how many real solutions a quadratic equation has. If \( \Delta > 0 \), there are two distinct real solutions. If \( \Delta = 0 \), there is one real solution. If \( \Delta < 0 \), there are no real solutions (but two complex ones).

Learn how to solve the quadratic equation 2x² + 3x - 9 = 0 using the quadratic formula. This detailed solution explains key algebraic concepts such as the discriminant and quadratic formula theorem. Suitable for students in Grades 8-10.

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