🎯 Learning Objectives
- What a quadratic equation looks like and why it has up to 2 solutions.
- When to factor, when to use the square root method, and when to use the quadratic formula.
- Use the quadratic formula with the correct coefficients.
- How the discriminant tells you the number of solutions.
- Strategies for SAT-style quadratics.
- Practice questions with worked solutions.
1. What Is a Quadratic Equation?
Standard form
a·x² + b·x + c = 0 (a ≠ 0)
A quadratic equation has degree 2. It can have 0, 1, or 2 real solutions, depending on the discriminant.
2. The 3 Solving Methods
Method | When to Use | Speed |
Factoring | When the quadratic factors easily | Convenient when factors are clear |
Square root | When there's no x term (only x² and constant) | Very fast |
Quadratic formula | When factoring fails | Always works |
3. Method 1: Factoring (Zero Product Property)
If two things multiply to zero, at least one of them must be zero. So if (x − 3)(x + 5) = 0, then x = 3 or x = −5.
📝 Example: Solve x² − 5x + 6 = 0
Factor: (x − 2)(x − 3) = 0
Set each factor to 0: x − 2 = 0 or x − 3 = 0
Solutions: x = 2 or x = 3
4. Method 2: Square Root Method
Use this when the equation has the form ax² = c (no x term). Solve for x², then take ± square root.
📝 Example: Solve 3x² = 27
Divide by 3: x² = 9
Take ± square root: x = ±3
5. Method 3: The Quadratic Formula
Quadratic Formula: a ≠ 0
x = (−b ± √(b² − 4ac)) / (2a)
To use it, identify a, b, and c from ax² + bx + c = 0, then plug in carefully.
6. The Discriminant: Counts Real Solutions
The 'discriminant': the part under the square root
Δ = b² − 4ac
❓ What is the value of b² − 4ac? | ||
Δ > 0 ▼ | Δ = 0 ▼ | |
TWO real solutions | ONE real solution (repeated) | |
💡 If Δ < 0
There are NO real solutions: the parabola never crosses the x-axis.
7. Strategies
⚡ STRATEGY 1: Always try factoring FIRST
Factoring is convenient when factors are easy to identify. The quadratic formula is also valid and is useful when factoring is less direct.
⚡ STRATEGY 2: Sum and Product of Roots
For ax² + bx + c = 0: sum of roots = −b/a, product of roots = c/a. The SAT often asks for one of these without ever requiring you to find the actual roots!
⚡ STRATEGY 3: Check the discriminant for 'how many solutions' questions
Don't solve. Just compute b² − 4ac. Done.
⚠️ Common Mistakes
- Forgetting to set the equation equal to zero before factoring.
- Forgetting the ± when taking a square root.
- Sign errors with −b in the quadratic formula.
- Dividing only one term when scaling the equation.
8. Summary
📌 Key Takeaways
- Standard form: ax² + bx + c = 0
- Methods: Factor → Square root → Quadratic formula.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a)
- Discriminant: b² − 4ac. Tells you 0, 1, or 2 real solutions.
- Sum of roots: −b/a
- Product of roots: c/a
9. Practice: 10 Questions
Q1. Solve x² = 49.
A) x = 7 B) x = ±7 C) x = ±49 D) x = 49
Hint: Square root method: ±√49 = ±7.
Q2. Solve x² − 6x + 8 = 0.
A) x = 2, 4 B) x = −2, −4 C) x = 1, 8 D) x = 6, 8
Hint: Factor: (x−2)(x−4).
Q3. What is the sum of the roots of 2x² − 10x + 8 = 0?
A) 4 B) 5 C) 8 D) 10
Hint: Sum = −b/a = 10/2 = 5.
Q4. What is the product of the roots of x² + 3x − 18 = 0?
A) −18 B) 18 C) −3 D) 3
Hint: Product = c/a = −18.
Q5. Solve 3x² − 12 = 0.
A) x = ±2 B) x = ±4 C) x = 12 D) x = 0
Hint: x² = 4 → x = ±2.
Q6. How many real solutions does x² + 4x + 5 = 0 have?
A) 0 B) 1 C) 2 D) infinite
Hint: Δ = 16 − 20 = −4 < 0 → 0 real solutions.
Q7. For what value of k does x² + kx + 9 = 0 have exactly one solution?
A) 3 B) 6 C) 9 D) ±6
Hint: Δ = k² − 36 = 0 → k = ±6.
Q8. Solve x² − x − 6 = 0.
A) x = 2, 3 B) x = −2, 3 C) x = 2, −3 D) x = −2, −3
Hint: Factor: (x−3)(x+2).
Q9. The roots of x² − 7x + c = 0 are 2 and 5. What is c?
A) 7 B) 10 C) 12 D) 14
Hint: Product of roots = c/a = c. 2 × 5 = 10.
Q10. Solve 2x² + 5x − 3 = 0 (use quadratic formula).
A) x = 1/2, −3 B) x = −1/2, 3 C) x = 3, −1/2 D) x = 1/2, 3
Hint: Discriminant = 25 + 24 = 49. x = (−5 ± 7)/4 = 1/2 or −3.
Answer Key & Worked Solutions
# | Answer | Reasoning |
Q1. | B) x = ±7 | Don't forget the ±. |
Q2. | A) x = 2, 4 | Factor (x−2)(x−4). |
Q3. | B) 5 | −b/a = 10/2 = 5. |
Q4. | A) −18 | c/a = −18. |
Q5. | A) x = ±2 | Square root method. |
Q6. | A) 0 | Discriminant negative. |
Q7. | D) ±6 | Δ = 0 ⇒ k² = 36. |
Q8. | A) x = 2, 3 | Wait: check signs. (x−3)(x+2)=0 ⇒ x=3 or x=−2. |
Q9. | B) 10 | Product of roots = 10. |
Q10. | A) x = 1/2, −3 | Quadratic formula. |