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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
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. The highest power of x is 2. The name 'quadratic' comes from the Latin 'quadratus' (square). Every quadratic has at most two roots - values of x that make the equation true. The graph of y = ax² + bx + c is always a parabola: opening upward (a > 0) or downward (a < 0).
Three main methods: (1) Quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a - works for all cases. (2) Factoring: rewrite as (px+q)(rx+s) = 0, works when roots are rational. (3) Completing the square: rewrite as a(x+h)² = k, then solve for x. The quadratic formula is the most universal - it always gives both roots whether real or complex. Choose factoring when the discriminant is a perfect square and an integer solution is apparent.
Discriminant Δ = b² − 4ac. Three cases: Δ > 0: two distinct real roots (parabola crosses x-axis twice). Δ = 0: one repeated root (parabola touches x-axis at exactly one point). Δ < 0: two complex conjugate roots (parabola does not cross x-axis). The discriminant is the value under the square root in the quadratic formula - its sign determines whether that square root is real or imaginary.
When Δ < 0, the square root involves i = √(−1), the imaginary unit. Complex roots always come in conjugate pairs: if x₁ = a + bi, then x₂ = a − bi. Example: x² + x + 1 = 0. Δ = 1−4 = −3. x = (−1 ± √(−3)) ÷ 2 = −1/2 ± (√3/2)i. These roots represent where the parabola would cross the x-axis if extended to the complex plane. Graphically, the parabola misses the real x-axis entirely.
The vertex is the turning point of the parabola - minimum if a > 0, maximum if a < 0. Coordinates: h = −b÷(2a), k = c − b²÷(4a), or equivalently k = f(h). In vertex form: y = a(x−h)² + k. The vertex lies exactly on the axis of symmetry. For x² − 6x + 5 = 0 (a=1, b=−6, c=5): h = 3, k = 5 − 9 = −4. Vertex at (3, −4).
Vieta's formulas relate the roots directly to the coefficients without solving. For roots x₁ and x₂: Sum of roots = x₁ + x₂ = −b÷a. Product of roots = x₁ × x₂ = c÷a. Example: x² − 5x + 6 = 0. Sum = 5÷1 = 5. Product = 6÷1 = 6. Check: 2+3=5 ✓ and 2×3=6 ✓. Vieta's formulas are useful for quickly checking solutions, constructing equations from given roots, and solving systems without the full quadratic formula.
Use factoring when: the discriminant is a non-negative perfect square (1, 4, 9, 25...) and the roots are rational integers or simple fractions. Factoring is faster and gives clean algebraic insight. Use the quadratic formula when: roots are irrational (Δ > 0 but not a perfect square), roots are complex (Δ < 0), coefficients are large or messy, or when you need a guaranteed method without guessing. The quadratic formula always works - factoring sometimes requires significant trial and error.
Completing the square rewrites ax² + bx + c in the form a(x+h)² + k. Steps for x² + 6x + 5: Move c to the right: x² + 6x = −5. Add (b/2)² = 9 to both sides: x² + 6x + 9 = 4. Factor left side: (x+3)² = 4. Solve: x+3 = ±2 → x = −1 or x = −5. This method reveals the vertex form directly: y = (x+3)² − 4, vertex at (−3, −4). It is also the algebraic derivation behind the quadratic formula.
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.
How this calculator works, and where the numbers come from
The Quadratic Equation Solver applies the standard formula for this calculation to the values you enter and updates the result as you type. The calculation itself happens in your browser, and the page explains the method so you can check any result by hand.
Please note: Results are provided for general information and are calculated from the values you enter.