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📐 Quadratic Equation Solver
Enter coefficients a, b, and c to instantly solve ax² + bx + c = 0. Get a full step-by-step quadratic formula walkthrough, discriminant analysis showing whether roots are real or complex, vertex and axis of symmetry, Vieta's formulas, and an interactive parabola graph - all in one place.
📐 Enter Coefficients of ax² + bx + c = 0
1x² + 0x + 0 = 0
Try:
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Discriminant (Δ = b²−4ac)
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Solutions
📊 Parabola Properties
Equation-
Parabola Opens-
Vertex (h, k)-
Axis of Symmetry-
Y-Intercept-
X-Intercept(s)-
Sum of Roots-
Product of Roots-
Discriminant (Δ)-
Nature of Roots-
📝 Step-by-Step Solution
🔲 Completing the Square
📈 Parabola Graph
Solve an equation in the Solver tab to see its graph here.
Parabola
Roots (x-intercepts)
Vertex
Y-intercept
📐 Quadratic Formula & Methods
The Quadratic Formula
For any equation ax² + bx + c = 0 (where a ≠ 0):
x = (−b ± √(b² − 4ac)) / (2a)
The ± gives TWO solutions:
x₁ = (−b + √(b²−4ac)) / (2a)
x₂ = (−b − √(b²−4ac)) / (2a)
The ± gives TWO solutions:
x₁ = (−b + √(b²−4ac)) / (2a)
x₂ = (−b − √(b²−4ac)) / (2a)
The Discriminant (Δ = b² − 4ac)
Δ > 0 → Two distinct real roots (parabola crosses x-axis twice)
Δ = 0 → One repeated real root (parabola touches x-axis once)
Δ < 0 → Two complex conjugate roots (parabola doesn't cross x-axis)
Δ = 0 → One repeated real root (parabola touches x-axis once)
Δ < 0 → Two complex conjugate roots (parabola doesn't cross x-axis)
Completing the Square
ax² + bx + c = 0
Step 1: x² + (b/a)x + c/a = 0
Step 2: x² + (b/a)x = -c/a
Step 3: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Step 4: (x + b/2a)² = (b²-4ac)/(4a²)
Step 5: x = -b/2a ± √(b²-4ac)/(2a)
Step 1: x² + (b/a)x + c/a = 0
Step 2: x² + (b/a)x = -c/a
Step 3: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Step 4: (x + b/2a)² = (b²-4ac)/(4a²)
Step 5: x = -b/2a ± √(b²-4ac)/(2a)
Factoring Method
Find two numbers m and n such that m×n = ac and m+n = b, then factor.
x² − 5x + 6 = 0
Find m×n = 6, m+n = −5: → m=−2, n=−3
(x − 2)(x − 3) = 0
x = 2 or x = 3
Find m×n = 6, m+n = −5: → m=−2, n=−3
(x − 2)(x − 3) = 0
x = 2 or x = 3
Vertex Form
Standard: y = ax² + bx + c
Vertex: y = a(x − h)² + k
Where vertex = (h, k):
h = −b / (2a)
k = c − b²/(4a) OR f(h)
Vertex: y = a(x − h)² + k
Where vertex = (h, k):
h = −b / (2a)
k = c − b²/(4a) OR f(h)
Vieta's Formulas
Relations between coefficients and roots (x₁, x₂):
Sum of roots: x₁ + x₂ = −b/a
Product of roots: x₁ × x₂ = c/a
Example: x²−5x+6=0 → roots 2,3
Sum: 2+3 = 5 = −(−5)/1 ✓
Product: 2×3 = 6 = 6/1 ✓
Product of roots: x₁ × x₂ = c/a
Example: x²−5x+6=0 → roots 2,3
Sum: 2+3 = 5 = −(−5)/1 ✓
Product: 2×3 = 6 = 6/1 ✓
❓ Frequently Asked Questions
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Quadratic Equation Solver - The Quadratic Formula, Discriminant & Parabola Explained
Quadratic equations appear throughout mathematics, physics, engineering, and finance - from projectile motion and lens optics to profit maximisation and signal processing. The quadratic formula solves any equation of the form ax² + bx + c = 0, and the discriminant tells you what kind of solution to expect before you even calculate it.
The quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. Example: x² − 5x + 6 = 0. a=1, b=−5, c=6. Δ = (−5)² − 4(1)(6) = 25 − 24 = 1. Roots: x = (5 ± 1) ÷ 2 → x = 3 and x = 2. Check: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓
The Discriminant - What It Tells You Before Solving
The discriminant Δ = b² − 4ac is the value under the square root in the quadratic formula. It predicts the type and number of roots without completing the full calculation:
Discriminant Cases
- Δ > 0: Two distinct real roots. The parabola crosses the x-axis at two points. If Δ is a perfect square, roots are rational (factoring works easily). If not, roots are irrational.
- Δ = 0: One repeated real root (double root). The parabola is tangent to the x-axis - touching at exactly one point (the vertex). x = −b÷(2a).
- Δ < 0: Two complex conjugate roots (no real roots). The parabola does not intersect the x-axis at all. Roots are a ± bi.
Quick Discriminant Examples
- x² − 5x + 6 = 0: Δ = 25 − 24 = 1 > 0 → two real roots: 2 and 3
- x² − 6x + 9 = 0: Δ = 36 − 36 = 0 → one root: x = 3
- x² + x + 1 = 0: Δ = 1 − 4 = −3 < 0 → complex roots: (−1 ± i√3) ÷ 2
- 2x² − 7x + 3 = 0: Δ = 49 − 24 = 25 > 0 → rational roots: 3 and 0.5
Three Methods to Solve Quadratic Equations
- Quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. Works for all quadratic equations without exception. The most reliable and general method. Derived by completing the square on the general form.
- Factoring: Express ax² + bx + c as a product of two linear factors. Only works cleanly when roots are rational. For x² − 5x + 6 = 0: find two numbers that multiply to 6 and add to −5 → −2 and −3. So (x−2)(x−3) = 0 → x = 2 or x = 3. Quick when it works, but requires trial and error or a rational discriminant.
- Completing the square: Rewrite as a(x + b/2a)² = b²/4a − c, then solve. This is how the quadratic formula is derived. It also reveals the vertex form y = a(x−h)² + k directly. Always works; particularly useful for graphing and calculus.
Key Properties of a Parabola - Vertex, Axis, and Intercepts
- Vertex: (h, k) where h = −b÷(2a) and k = c − b²÷(4a). The turning point - minimum if a > 0, maximum if a < 0.
- Axis of symmetry: The vertical line x = −b÷(2a) = h. The parabola is a perfect mirror image on both sides.
- y-intercept: Set x = 0 → y = c. Always a single point.
- x-intercepts (roots): Where the parabola crosses y = 0. Found by the quadratic formula. 0, 1, or 2 real x-intercepts depending on the discriminant.
- Parabola opens up: if a > 0. Parabola opens down: if a < 0.
- Vieta's formulas: For roots x₁ and x₂: x₁ + x₂ = −b÷a and x₁ × x₂ = c÷a. Useful for checking solutions without substitution.