What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in the standard form ax² + bx + c = 0, where a ≠ 0. The variable x represents the unknown value we solve for, while a, b, and c are known coefficients. Quadratic equations appear everywhere in mathematics, physics, engineering, economics, and computer science.
Every quadratic equation can have zero, one, or two real solutions (called roots), depending on the value of the discriminant (Δ = b² - 4ac). Our quadratic calculator online instantly identifies the root type and computes exact or approximate solutions with full working shown.
This quadratic solver uses the universally applicable quadratic formula, derived by completing the square:
How to Use This Quadratic Calculator
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Enter Coefficients a, b, c
Input the coefficients from your equation ax² + bx + c = 0. For example, for 2x² - 5x + 3 = 0, enter a=2, b=-5, c=3. Negative values and decimals are fully supported. The calculator validates that a ≠ 0.
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Click "Solve Equation"
Press the calculate button. Our engine instantly computes the discriminant, determines root type, applies the quadratic formula, and calculates vertex coordinates using optimized mathematical algorithms.
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Review Results & Steps
View roots, discriminant, vertex, and axis of symmetry in the results grid. Scroll down to see a complete step-by-step breakdown showing substitution into the quadratic formula and simplification.
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Copy, Share, or Reset
Use Copy to export the full solution to your clipboard, Share to send via messaging apps, or Reset to solve a new equation. Your privacy is protected — no data is stored or transmitted.
Worked Examples
See how this quadratic solver calculator handles different equation types:
Example 1: Two Real Roots
Equation: x² - 5x + 6 = 0 (a=1, b=-5, c=6)
Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1 (positive → two real roots)
Roots: x = (5 ± √1) / 2 = (5 ± 1) / 2 → x₁ = 3, x₂ = 2
Vertex: h = -(-5)/(2×1) = 2.5, k = f(2.5) = -0.25 → (2.5, -0.25)
Example 2: One Repeated Root
Equation: x² - 4x + 4 = 0 (a=1, b=-4, c=4)
Discriminant: Δ = (-4)² - 4(1)(4) = 16 - 16 = 0 (zero → one repeated root)
Root: x = (4 ± √0) / 2 = 4/2 → x = 2 (multiplicity 2)
Vertex: h = 2, k = 0 → (2, 0) (vertex touches x-axis)
Example 3: Complex Roots
Equation: x² + 2x + 5 = 0 (a=1, b=2, c=5)
Discriminant: Δ = (2)² - 4(1)(5) = 4 - 20 = -16 (negative → complex roots)
Roots: x = (-2 ± √-16) / 2 = (-2 ± 4i) / 2 → x₁ = -1 + 2i, x₂ = -1 - 2i
Vertex: h = -1, k = 4 → (-1, 4) (parabola opens upward, no x-intercepts)
Pro Tips for Quadratic Equations
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Check the Discriminant First
Before solving, calculate Δ = b² - 4ac. This tells you immediately whether to expect real or complex roots, saving time and guiding your approach to graphing or further analysis.
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Vertex Form for Graphing
Convert to vertex form y = a(x-h)² + k to easily sketch parabolas. The vertex (h,k) is the turning point, and 'a' determines direction (upward if a>0, downward if a<0) and width.
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Verify Your Solutions
Always substitute your roots back into the original equation to verify. For x=3 in x²-5x+6=0: 3²-5(3)+6 = 9-15+6 = 0 ✓. This catches calculation errors instantly.
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When Factoring Works
If the quadratic factors nicely (e.g., x²-5x+6 = (x-2)(x-3)), factoring is faster than the formula. Use this calculator to check your factoring work or when equations don't factor easily.