Quadratic Equations
Grade 9 · Math · Study Guide
Solve quadratic equations by factoring, square roots, and the quadratic formula, and understand the discriminant.
A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. The graph of a quadratic is a U-shaped curve called a parabola. The solutions (also called roots or zeros) are the x-values where the parabola crosses the x-axis.
Factoring is the fastest method when it works. Set the equation equal to zero, factor, then apply the Zero Product Property: if (x − 3)(x + 2) = 0, then x = 3 or x = −2. For simple equations like x² = 49, use square roots: x = ±7.
The quadratic formula works for any quadratic: x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant. If it's positive, there are two real solutions; if zero, exactly one; if negative, no real solutions.
| Discriminant (b²−4ac) | Number of Real Solutions | Graph Behavior |
|---|---|---|
| Positive | 2 real solutions | Parabola crosses x-axis twice |
| Zero | 1 real solution | Parabola touches x-axis once (vertex) |
| Negative | No real solutions | Parabola does not cross x-axis |
Key terms
- Quadratic equation
- An equation of the form ax² + bx + c = 0 where a ≠ 0
- Roots/zeros
- The solutions to a quadratic equation; x-intercepts of the parabola
- Zero Product Property
- If ab = 0, then a = 0 or b = 0 (or both)
- Quadratic formula
- x = (−b ± √(b²−4ac)) / 2a — solves any quadratic
- Discriminant
- b² − 4ac; determines the number and type of solutions
- Parabola
- The U-shaped graph of a quadratic function
Flashcards
- Standard form of a quadratic
- ax² + bx + c = 0, where a ≠ 0
- Zero Product Property
- If ab = 0, then a = 0 or b = 0
- Quadratic formula
- x = (−b ± √(b²−4ac)) / 2a
- Discriminant
- b² − 4ac; positive → 2 roots, zero → 1 root, negative → no real roots
- Parabola
- The graph of a quadratic function — U-shaped, opening up or down
- Roots of a quadratic
- The x-values where the parabola intersects the x-axis
Practice questions
Solve by factoring: x² + 5x + 6 = 0
- x = 2 or x = 3
- x = −2 or x = −3
- x = 1 or x = 6
- x = −1 or x = −6
Show answer
Answer: x = −2 or x = −3
Factor: (x + 2)(x + 3) = 0 → x = −2 or x = −3.
A quadratic equation can have exactly two real solutions.
Show answer
Answer: True
When the discriminant is positive, the quadratic formula yields two distinct real solutions.
The discriminant of 2x² − 4x + 2 = 0 is b² − 4ac = ___.
Show answer
Answer: 0
b² − 4ac = (−4)² − 4(2)(2) = 16 − 16 = 0.
Solve: x² = 25
- x = 5
- x = −5
- x = ±5
- x = ±25
Show answer
Answer: x = ±5
Take square roots of both sides: x = √25 = ±5.
Using the quadratic formula, solve x² − 5x + 6 = 0
- x = 2 or x = 3
- x = −2 or x = 3
- x = 1 or x = 6
- x = 2 or x = −3
Show answer
Answer: x = 2 or x = 3
a=1, b=−5, c=6; discriminant = 25−24 = 1; x = (5±1)/2 = 3 or 2.
Fun facts
- Quadratic equations model projectile motion — the path of a thrown ball or a rocket follows a parabola.
- The word 'quadratic' comes from the Latin 'quadratus' meaning square — because the variable is squared.
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