The worksheet appears to involve solving quadratic equations, finding discriminants, and determining the number of solutions for given equations. Let’s analyze the visible problems step-by-step.

Part 4: Finding the Discriminant & Number of Solutions

1. Formula for the Discriminant: $\Delta = b^2 - 4ac$ Where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.
   The number of solutions depends on the discriminant:
   • If $\Delta > 0$: Two distinct real solutions.
   • If $\Delta = 0$: One real solution (repeated root).
   • If $\Delta < 0$: No real solutions.

Problem Analysis:

7. Equation: $3x^2 + 11x + 12 = 0$

• Coefficients: $a = 3$, $b = 11$, $c = 12$
• Discriminant: $\Delta = b^2 - 4ac = 11^2 - 4(3)(12) = 121 - 144 = -23$
• Since $\Delta < 0$, there are 0 real solutions.

8. Equation: $-9x^2 + 3x + 6 = 0$

• Coefficients: $a = -9$, $b = 3$, $c = 6$
• Discriminant: $\Delta = b^2 - 4ac = 3^2 - 4(-9)(6) = 9 + 216 = 225$
• Since $\Delta > 0$, there are 2 real solutions.

9. Equation: $x^2 + 6x = -9$

Rewrite as: $x^2 + 6x + 9 = 0$ (adding 9 to both sides).

• Coefficients: $a = 1$, $b = 6$, $c = 9$
• Discriminant: $\Delta = b^2 - 4ac = 6^2 - 4(1)(9) = 36 - 36 = 0$
• Since $\Delta = 0$, there is 1 real solution.

Summary of Solutions

• 7: Discriminant = $-23$, Number of solutions = 0
• 8: Discriminant = $225$, Number of solutions = 2
• 9: Discriminant = $0$, Number of solutions = 1

Do you need further details or help with other problems on this worksheet?

5 Related Questions

1. How is the discriminant related to the roots of a quadratic equation geometrically?
2. Can a quadratic equation have exactly one real and one imaginary root?
3. What is the significance of the discriminant being zero in physics-related problems?
4. How does factoring a quadratic differ from using the quadratic formula?
5. What are real-world examples of quadratic equations with no real solutions?

Tip: When calculating the discriminant, always double-check the signs of $a$, $b$, and $c$ to avoid mistakes.