Use the discriminant to determine whether the quadratic equation has two unequal real solutions, a repeated real solution, or no real solution, without solving the equation. \( 2 x^{2}+7 x-9=0 \) Which of the following correctly describes the solutions to the given equation? A. A repeated real solution B. No real solution C. Two unequal real solutions

To determine the nature of the solutions for the quadratic equation \( 2x^{2} + 7x - 9 = 0 \) without solving it, we can use the **discriminant** method. ### Step 1: Identify the coefficients A quadratic equation is generally in the form: \[ ax^{2} + bx + c = 0 \] For the given equation: - \( a = 2 \) - \( b = 7 \) - \( c = -9 \) ### Step 2: Calculate the Discriminant The discriminant (\( D \)) is given by the formula: \[ D = b^{2} - 4ac \] Plugging in the values: \[ D = (7)^{2} - 4 \times 2 \times (-9) \] \[ D = 49 - (-72) \] \[ D = 49 + 72 \] \[ D = 121 \] ### Step 3: Analyze the Discriminant - **If \( D > 0 \)**: The equation has **two unequal real solutions**. - **If \( D = 0 \)**: The equation has **one repeated real solution**. - **If \( D < 0 \)**: The equation has **no real solution** (but two complex solutions). In this case: \[ D = 121 > 0 \] ### Conclusion Since the discriminant is positive, the quadratic equation \( 2x^{2} + 7x - 9 = 0 \) has **two unequal real solutions**. **Answer:** **C. Two unequal real solutions**