Solve quadratic equations ax²+bx+c=0 using the quadratic formula. Calculate discriminant, find real and complex roots, vertex coordinates, and analyze parabola properties with step-by-step solutions.
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The quadratic formula is a powerful mathematical tool that provides solutions (roots) to any quadratic equation, making it essential for algebra, calculus, physics, and engineering applications.
Understanding quadratic equations is fundamental to mathematics and science, as they appear in countless real-world scenarios including projectile motion, optimization problems, area calculations, and parabolic shapes in architecture and engineering.
Where: a, b, c are coefficients and a ≠ 0
The discriminant determines the nature of the roots
| Equation | a | b | c | Discriminant | Root Type | Solutions |
|---|---|---|---|---|---|---|
| x² - 5x + 6 = 0 | 1 | -5 | 6 | 1 | Two Real | x = 2, 3 |
| x² - 4x + 4 = 0 | 1 | -4 | 4 | 0 | One Real | x = 2 |
| x² + x + 1 = 0 | 1 | 1 | 1 | -3 | Complex | x = -1/2 ± i√3/2 |
| 2x² - 8x + 6 = 0 | 2 | -8 | 6 | 16 | Two Real | x = 1, 3 |
| x² - 6x + 9 = 0 | 1 | -6 | 9 | 0 | One Real | x = 3 |
| 3x² + 2x + 1 = 0 | 3 | 2 | 1 | -8 | Complex | x = -1/3 ± i√2/3 |
Vertex Form: A quadratic function can be written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex coordinates are h = -b/(2a) and k = f(h).
Axis of Symmetry: The vertical line x = -b/(2a) divides the parabola into two symmetric halves. This line passes through the vertex.
Direction of Opening: If a > 0, the parabola opens upward (U-shaped). If a < 0, the parabola opens downward (∩-shaped).
Y-intercept: The point where the parabola crosses the y-axis, which occurs at (0, c) when x = 0.
The quadratic formula is derived by completing the square for the general quadratic equation ax² + bx + c = 0:
Step 1: Start with ax² + bx + c = 0
Step 2: Divide by a: x² + (b/a)x + c/a = 0
Step 3: Move constant term: x² + (b/a)x = -c/a
Step 4: Complete the square: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Step 5: Factor left side: (x + b/2a)² = (b² - 4ac)/(4a²)
Step 6: Take square root: x + b/2a = ±√(b² - 4ac)/(2a)
Step 7: Solve for x: x = (-b ± √(b² - 4ac))/(2a)
Projectile Motion: The height of a projectile follows a quadratic equation h(t) = -½gt² + v₀t + h₀, where solving for h(t) = 0 gives the time when the projectile hits the ground.
Area Optimization: Many optimization problems involving area result in quadratic equations. For example, finding dimensions that maximize the area of a rectangle with a fixed perimeter.
Profit Maximization: Business profit functions are often quadratic, where the maximum profit occurs at the vertex of the parabola.
Engineering and Physics: Quadratic equations appear in electrical circuits, mechanical systems, wave equations, and structural analysis.
Factoring: When the quadratic can be factored into (px + q)(rx + s) = 0, the solutions are x = -q/p and x = -s/r. This method works well when roots are rational.
Completing the Square: Converting ax² + bx + c = 0 to the form a(x - h)² + k = 0. This method is useful for finding vertex form and understanding parabola properties.
Graphing: Plotting the function y = ax² + bx + c and finding where it intersects the x-axis. This visual method helps understand the behavior of the function.
Using Technology: Graphing calculators, computer software, and online tools can quickly solve quadratic equations and provide visual representations.
When a = 0: The equation becomes linear (bx + c = 0), which is not actually quadratic. Always ensure a ≠ 0 for quadratic equations.
Rational vs Irrational Roots: If the discriminant is a perfect square, the roots are rational. If it's positive but not a perfect square, the roots are irrational.
Integer Solutions: Quadratic equations with integer coefficients may have integer solutions when the discriminant is a perfect square and certain divisibility conditions are met.