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Quadratic Equation

ax²+bx+c=0. Roots: x=(−b±√(b²−4ac))/(2a). Discriminant Δ=b²−4ac: >0 two real, =0 one (double), <0 two complex. Vertex (h,k): h=−b/(2a), k=c−b²/(4a). Vieta: sum=−b/a, product=c/a.

Concept Fundamentals
x=(−b±√Δ)/(2a)
Formula
Δ=b²−4ac
Discriminant
h=−b/(2a)
Vertex
sum=−b/a, prod=c/a
Vieta
Solve QuadraticFormula, completing square, factoring

Why This Mathematical Concept Matters

Why: Quadratics model projectiles, optimization, and many natural phenomena. Discriminant reveals root nature. Vertex gives max/min. Vieta links coefficients to roots. Foundational for calculus and physics.

How: Formula: compute Δ=b²−4ac, then x=(−b±√Δ)/(2a). Completing square: ax²+bx+c=a(x+b/(2a))²+(c−b²/(4a)). Factoring: find r,s with r+s=−b/a, rs=c/a, then (x−r)(x−s)=0.

  • ●Δ>0: two distinct real roots. Δ=0: one repeated root. Δ<0: two complex conjugates.
  • ●Vertex at x=−b/(2a) — axis of symmetry. Parabola opens up if a>0, down if a<0.
  • ●Vieta's formulas: x₁+x₂=−b/a, x₁x₂=c/a — no need to compute roots for sum/product.
Sources:Khan Academy — Quadratic EquationsWolfram MathWorld — Quadratic Equation

📐 Examples — Click to Load

Coefficients: ax² + bx + c = 0

quadratic_solver.sh
CALCULATED
x₁
3
x₂
2
Δ (discriminant)
1
Vertex (h, k)
(2.5, -0.25)
Axis of symmetry
x = 2.5
Vieta's: sum, product
5, 6
Factored form
(x - 3)(x - 2)
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Coefficients & Discriminant

Root Type

Given: x² - 5x + 6 = 0
Coefficients: a = 1, b = -5, c = 6
Discriminant: Δ = b² - 4ac = -5² - 4(1)(6) = 25 - 24 = 1
Δ > 0: Two distinct real roots
x₁ = (-b + √Δ)/(2a) = (5 + 1)/2 = 3
x₂ = (-b - √Δ)/(2a) = (5 - 1)/2 = 2
Vertex: (h, k) = (2.5, -0.25)
Axis of symmetry: x = 2.5
Vieta's: x₁ + x₂ = 5, x₁·x₂ = 6

⚠️For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📐

x²−5x+6=0: roots 2 and 3. Sum=5, product=6. Vieta checks out.

— Example

◻

Δ=0 means parabola touches x-axis at one point (vertex).

— Discriminant

📋 Quadratic Formula Derivation

Starting from ax² + bx + c = 0, divide by a: x² + (b/a)x + c/a = 0. Complete the square: x² + (b/a)x + (b/2a)² = (b/2a)² - c/a. Factor left: (x + b/2a)² = (b² - 4ac)/(4a²). Take square roots: x + b/2a = ±√(b²-4ac)/(2a). Thus x = (-b ± √(b²-4ac))/(2a).

📊 Discriminant Classification

Δ = b² - 4acRootsParabola
Δ > 0Two distinct real rootsCrosses x-axis twice
Δ = 0One repeated rootTouches x-axis at vertex
Δ < 0Two complex conjugate rootsNo x-intercepts

🔢 Vieta's Formulas

For ax² + bx + c = 0 with roots x₁ and x₂: x₁ + x₂ = -b/a and x₁ · x₂ = c/a. These relate coefficients to roots without computing the roots explicitly.

📐 Parabola Geometry

Vertex form: y = a(x - h)² + k where h = -b/(2a) and k = c - b²/(4a). The axis of symmetry is x = h. If a > 0 the parabola opens up (minimum at vertex); if a < 0 it opens down (maximum).

⚠️ Common Mistakes

  • Forgetting a ≠ 0 — if a = 0 it's linear, not quadratic.
  • Sign errors: -b means the opposite of b; 4ac is always positive when a and c have the same sign.
  • When Δ < 0, √Δ = i√|Δ|; the roots are complex conjugates.

❓ FAQ

What is the quadratic formula?

x = (-b ± √(b²-4ac))/(2a) solves any ax²+bx+c=0. The ± gives two roots.

When do I get complex roots?

When the discriminant Δ = b²-4ac is negative. The roots are a ± bi where a and b are real.

What is the vertex of a parabola?

The vertex is at (h,k) where h = -b/(2a) and k = c - b²/(4a). It is the minimum (a>0) or maximum (a<0).

What are Vieta's formulas?

For roots x₁ and x₂: x₁+x₂ = -b/a and x₁·x₂ = c/a. Useful to check answers or find coefficients from roots.

When should I use factoring vs the formula?

Factoring is faster when roots are nice integers. Use the formula when factoring is hard or roots are irrational.

📝 Worked Examples

Example 1: x²-5x+6=0 — a=1, b=-5, c=6. Δ = 25-24 = 1. x = (5±1)/2 → x₁=3, x₂=2. Factored: (x-2)(x-3).
Example 2: x²+4=0 — a=1, b=0, c=4. Δ = -16. x = ±(4i)/2 = ±2i. Complex conjugate roots.
Example 3: x²-6x+9=0 — Δ = 36-36 = 0. One root: x = 3. Perfect square: (x-3)².

🎯 Vertex Form Conversion

To convert ax²+bx+c to vertex form a(x-h)²+k: Complete the square. Factor out a from the first two terms, add and subtract (b/2a)² inside, then simplify. The result is a(x + b/2a)² + (c - b²/4a), so h = -b/2a and k = c - b²/4a.

📌 Summary

The quadratic equation ax²+bx+c=0 has solutions given by the quadratic formula. The discriminant determines whether roots are real (Δ≥0) or complex (Δ<0). The vertex (h,k) gives the parabola's turning point. Vieta's formulas relate sum and product of roots to coefficients. Use this calculator for any quadratic — from simple factoring to projectile motion and golden ratio problems.

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