Question

Complete the following activity to determine the nature of the roots of the quadratic equation \(x^2 + 2x - 9 = 0\):

Hint:

Use the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of roots: \begin{itemize} \item \(\Delta > 0\): Real and unequal roots \item \(\Delta = 0\): Real and equal roots \item \(\Delta < 0\): Imaginary roots \end{itemize}

Answer 1

Step 1: Compare the given equation with the standard quadratic form. 
Given equation: \[ x^2 + 2x - 9 = 0 \] Standard form: \[ ax^2 + bx + c = 0 \] By comparison, we get: \[ a = 1, \quad b = 2, \quad c = -9 \] Step 2: Recall the discriminant formula. 
\[ \Delta = b^2 - 4ac \] Step 3: Substitute the values. 
\[ \Delta = (2)^2 - 4(1)(-9) \] Step 4: Simplify. 
\[ \Delta = 4 + 36 = 40 \] Step 5: Interpret the discriminant. 
Since \(\Delta = 40>0\), the discriminant is positive. 
Step 6: Conclusion. 
If \(b^2 - 4ac>0\), then the roots of the quadratic equation are real and unequal. 
Final Answer: \[ \boxed{\text{The roots of the equation are real and unequal.}} \]