Quadratic Equation Calculator
Solve quadratic equations (ax² + bx + c = 0) and find roots, discriminant, vertex, and more.
Enter Coefficients
ax² + bx + c = 0
Your Equation:
x² - 5x + 6 = 0
x = (-b ± √(b² - 4ac)) / 2a
Solutions (Roots)
x₁ = 3
x₂ = 2
Two distinct real roots
ΔDiscriminant
1
📍Vertex
(2.5, -0.25)
Parabola Properties
Axis of Symmetryx = 2.5
Y-Intercept(0, 6)
OpensUpward ∪
Discriminant (b² - 4ac)1
Vieta's Formulas
Sum of roots (x₁ + x₂)5
Product of roots (x₁ × x₂)6
Interpretation
The parabola crosses the x-axis at two points: x = 3 and x = 2
The vertex is at (2.5, -0.25), which is the minimum point.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. These equations are fundamental in algebra and appear throughout mathematics, physics, engineering, and economics.
Standard Form: ax² + bx + c = 0
- a = coefficient of x² (must not be zero)
- b = coefficient of x
- c = constant term
Key Properties:
- Always has exactly 2 solutions (roots), which may be real or complex
- The graph is always a parabola
- If a > 0, parabola opens upward (U-shape)
- If a < 0, parabola opens downward (∩-shape)
- The vertex is the maximum or minimum point
Other Forms:
- Vertex Form: a(x - h)² + k, where (h, k) is the vertex
- Factored Form: a(x - r₁)(x - r₂), where r₁ and r₂ are the roots
The Quadratic Formula
The quadratic formula provides a direct method to solve any quadratic equation:
Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
Where:
• a, b, c are coefficients from ax² + bx + c = 0
• ± means there are two solutions
• The expression under √ is the discriminant (D)
Discriminant: D = b² - 4ac
• D > 0: Two distinct real solutions
• D = 0: One repeated real solution
• D < 0: Two complex conjugate solutions
Where:
- a= Coefficient of x² (cannot be zero)
- b= Coefficient of x
- c= Constant term
- D= Discriminant (b² - 4ac)
Understanding the Discriminant
The discriminant (D = b² - 4ac) determines the nature of the solutions:
| Discriminant | Nature of Roots | Graph Intersection | Example |
|---|---|---|---|
| D > 0 (Positive) | Two distinct real roots | Parabola crosses x-axis at 2 points | x² - 5x + 6 = 0 → x = 2, 3 |
| D = 0 (Zero) | One repeated real root | Parabola touches x-axis at 1 point | x² - 4x + 4 = 0 → x = 2 |
| D < 0 (Negative) | Two complex conjugate roots | Parabola doesn't cross x-axis | x² + 1 = 0 → x = ±i |
Perfect Square Discriminant: When D is a perfect square (like 0, 1, 4, 9, 16...), the roots are rational numbers and the equation can be factored nicely.
Methods for Solving Quadratic Equations
There are several ways to solve quadratic equations:
1. Factoring (When Possible)
- Express ax² + bx + c as a(x - r₁)(x - r₂)
- Set each factor equal to zero
- Works when roots are rational numbers
- Example: x² - 5x + 6 = (x - 2)(x - 3) = 0, so x = 2 or x = 3
2. Quadratic Formula (Always Works)
- Plug a, b, c into x = (-b ± √(b² - 4ac)) / 2a
- Works for all quadratic equations
- Handles irrational and complex roots
3. Completing the Square
- Rewrite in the form (x + p)² = q
- Take square root of both sides: x + p = ±√q
- Useful for deriving the quadratic formula
4. Graphing
- Plot y = ax² + bx + c
- Find x-intercepts (where y = 0)
- Gives visual approximation of roots
Vertex and Parabola Properties
The vertex is the highest or lowest point on the parabola:
Vertex Formulas
Vertex x-coordinate: x = -b / 2a
Vertex y-coordinate: y = f(-b/2a) = c - b²/4a
Axis of Symmetry: x = -b / 2a
Vertex Form: y = a(x - h)² + k
where (h, k) is the vertex
Where:
- h= x-coordinate of vertex = -b/2a
- k= y-coordinate of vertex
- a= Determines width and direction of parabola
How to Use This Quadratic Calculator
Our calculator solves any quadratic equation step by step:
- Enter Coefficients: Input values for a, b, and c from ax² + bx + c = 0
- Click Calculate: The calculator applies the quadratic formula
- View Results:
- Both solutions (x₁ and x₂)
- Discriminant value and what it means
- Vertex coordinates
- Axis of symmetry
- Factored form (if applicable)
Tips:
- For equations like x² - 4 = 0, enter a=1, b=0, c=-4
- Remember: a cannot be zero (that would be linear, not quadratic)
- Complex solutions are shown in a + bi form
Real-World Applications of Quadratic Equations
Quadratic equations model many real-world phenomena:
Physics - Projectile Motion:
- Height of a thrown ball: h(t) = -16t² + v₀t + h₀
- Finding when object hits ground (h = 0)
- Maximum height occurs at vertex
Business - Profit Optimization:
- Profit = Revenue - Cost often forms a parabola
- Maximum profit at vertex
- Break-even points where profit = 0
Engineering:
- Parabolic reflectors (satellite dishes, headlights)
- Arch bridge design
- Suspension cable shapes
Area Problems:
- Finding dimensions when area is known
- Maximizing area with fixed perimeter
Finance:
- Compound interest calculations
- Present value problems
Worked Examples
Solve Using Quadratic Formula
Problem:
Solve x² - 5x + 6 = 0
Solution Steps:
- 1Identify coefficients: a = 1, b = -5, c = 6
- 2Calculate discriminant: D = (-5)² - 4(1)(6) = 25 - 24 = 1
- 3D > 0, so there are two distinct real roots
- 4Apply formula: x = (5 ± √1) / 2
- 5x = (5 + 1)/2 = 3 or x = (5 - 1)/2 = 2
- 6Verify: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓
Result:
x = 3 or x = 2
Complex Roots Example
Problem:
Solve x² + 2x + 5 = 0
Solution Steps:
- 1Identify coefficients: a = 1, b = 2, c = 5
- 2Calculate discriminant: D = (2)² - 4(1)(5) = 4 - 20 = -16
- 3D < 0, so roots are complex
- 4Apply formula: x = (-2 ± √(-16)) / 2
- 5√(-16) = 4i (where i = √-1)
- 6x = (-2 ± 4i) / 2 = -1 ± 2i
Result:
x = -1 + 2i or x = -1 - 2i
Repeated Root Example
Problem:
Solve x² - 6x + 9 = 0
Solution Steps:
- 1Identify coefficients: a = 1, b = -6, c = 9
- 2Calculate discriminant: D = (-6)² - 4(1)(9) = 36 - 36 = 0
- 3D = 0, so there is one repeated root
- 4Apply formula: x = (6 ± √0) / 2 = 6/2 = 3
- 5This is a perfect square: x² - 6x + 9 = (x - 3)²
- 6Vertex touches x-axis at x = 3
Result:
x = 3 (double root)
Tips & Best Practices
- ✓Always check by substituting your solutions back into the original equation
- ✓Factoring is fastest when possible - look for patterns first
- ✓The vertex is at x = -b/2a - this gives the maximum or minimum y-value
- ✓If the discriminant is a perfect square, the roots are rational (factorable)
- ✓Remember: a cannot be zero, or it's not a quadratic equation
- ✓Complex roots always come in conjugate pairs (a+bi and a-bi)
- ✓Use the sum and product of roots to verify: sum = -b/a, product = c/a
Frequently Asked Questions
When the discriminant (b² - 4ac) is negative, the equation has no real solutions. Instead, it has two complex conjugate solutions in the form a + bi and a - bi, where i = √(-1). For example, x² + 4 = 0 has solutions x = ±2i. These complex roots mean the parabola doesn't intersect the x-axis.
Start with factoring if the coefficients are small integers - look for two numbers that multiply to give ac and add to give b. If factoring isn't obvious, use the quadratic formula - it always works. Completing the square is useful when converting to vertex form or deriving formulas. Graphing gives a visual check of your solutions.
Vertex form is y = a(x - h)² + k, where (h, k) is the vertex. It's useful when you need to quickly identify the maximum/minimum value (k) and where it occurs (x = h). To convert from standard form: h = -b/2a, then k = f(h). This form is especially useful in optimization problems.
Yes, when the discriminant equals zero. In this case, both solutions from the quadratic formula are the same number (a 'double root' or 'repeated root'). Graphically, the parabola touches the x-axis at exactly one point (the vertex). Example: x² - 4x + 4 = 0 has only x = 2 as a solution.
Substitute each solution back into the original equation. If the equation equals zero, the solution is correct. For example, if x = 3 is a solution to x² - 5x + 6 = 0, check: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓. Also, the sum of roots equals -b/a and the product equals c/a.
For ax² + bx + c = 0 with roots r₁ and r₂: Sum of roots: r₁ + r₂ = -b/a. Product of roots: r₁ × r₂ = c/a. These relationships (Vieta's formulas) let you check your answers or find information about roots without solving the equation completely.
Sources & References
Last updated: 2026-01-22