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Quadratic Equation Solver

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1x² + -5x + 6 = 0

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Algebra & Polynomial Math

Quadratic Equation Solver — Complete Method Guide

Solve ax² + bx + c = 0 by quadratic formula, factoring, or completing the square. Interpret real and complex roots, the discriminant, parabola geometry, and Vieta's formulas.

Quadratic Formula

x = (−b ± √D) / 2a

Discriminant

D = b² − 4ac

Sum of Roots

r₁ + r₂ = −b/a

Product of Roots

r₁ × r₂ = c/a

✓ Reviewed by the CalculatorApp Mathematics & Algebra Team

What Is a Quadratic Equation?

A quadratic equation has the standard form ax² + bx + c = 0, where a ≠ 0. Its graph is a parabola: opening upward if a > 0, downward if a < 0. The solutions (roots) are where the parabola crosses the x-axis — real if D ≥ 0, complex (conjugate pair) if D < 0.

Quadratic equations model projectile trajectories, profit maximization, circuit resonance, structural deflection, and lens optics. The quadratic formula — derived by completing the square — always yields both roots regardless of the discriminant sign.

Solution Methods

Quadratic Formula

x = (−b ± √(b²−4ac)) / 2a

Works for all quadratics

Factoring

(x − r₁)(x − r₂) = 0

Fastest when roots are integers

Completing the Square

(x + b/2a)² = (b²−4ac)/4a²

Derives the formula itself

Vieta's Formulas

r₁+r₂=−b/a, r₁r₂=c/a

Relate roots to coefficients

Discriminant Analysis

Discriminant DRoot TypeGraph BehaviorAction
D > 02 distinct real rootsParabola crosses x-axis twiceCompute both: x = (−b ± √D) / 2a
D = 01 repeated real rootParabola touches x-axis at vertexx = −b / 2a (single root)
D < 02 complex conjugate rootsParabola does not cross x-axisx = (−b ± i√|D|) / 2a
a = 0Not quadratic — linear equationStraight line (bx + c = 0)x = −c / b if b ≠ 0

History of Quadratic Equations

~2000 BC — Babylonian

Solved quadratic-type area problems numerically, without symbolic algebra.

~600 BC — Indian Vedic

Brahmagupta describes rules for solving quadratics including negative solutions.

830 AD — Al-Khwarizmi

Formalizes algebraic solution methods in Al-Kitab al-mukhtasar; the word algebra derives from this work.

16th century — European

Cardano and Ferrari extend to cubics and quartics; Vieta introduces symbolic notation.

17th century — Descartes

Introduces the Cartesian plane — quadratics become parabolas visualized geometrically.

Modern era — Applied

Quadratics model projectile motion, profit optimization, electrical resonance, and optics.

Key Resources & Research

External link

Wolfram MathWorld — Quadratic

Deep mathematical treatment of quadratic equations, discriminants, and root classification.

External link

Khan Academy — Quadratics

Free lessons on factoring, completing the square, and the quadratic formula.

External link

NIST — Algebra Reference

Authoritative mathematical functions and polynomial reference data.

External link

Paul's Online Math Notes

Comprehensive algebra and calculus notes widely used in universities.

External link

MIT OpenCourseWare — Algebra

Free lecture notes and problem sets covering polynomials and complex numbers.

External link

3Blue1Brown — Essence of Algebra

Visual explanations of algebraic concepts including parabolas and roots.

Myths vs Facts

❌ Myth: A quadratic equation always has two different real roots.

✅ Fact: When D = 0 there is exactly one repeated root; when D < 0, both roots are complex (not real).

❌ Myth: You cannot take the square root of a negative number.

✅ Fact: You can — using imaginary numbers. √(−4) = 2i, where i = √(−1). The roots are complex conjugates: a ± bi.

❌ Myth: Factoring is always possible for any quadratic.

✅ Fact: Factoring over the integers is only possible when the discriminant is a perfect square. Otherwise, use the quadratic formula.

❌ Myth: The coefficients a, b, c must all be integers.

✅ Fact: The quadratic formula works for any real (or complex) coefficients. Decimal or fractional coefficients are fine.

Frequently Asked Questions (12)

What is the quadratic formula?+
The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / (2a). It is derived by completing the square on the general form. The ± produces two solutions. If the discriminant D = b²−4ac is negative, the roots are complex conjugates: x = −b/(2a) ± i√(|D|)/(2a).
What is the discriminant and what does it tell you?+
D = b² − 4ac. If D > 0: two distinct real roots (parabola crosses x-axis twice). If D = 0: one repeated real root (parabola tangent to x-axis). If D < 0: two complex conjugate roots (parabola does not touch x-axis). The discriminant determines the nature of roots before computing them.
How do I factor a quadratic equation?+
For x² + bx + c, find two numbers that multiply to c and add to b. Example: x² + 5x + 6 = (x+2)(x+3) because 2×3=6 and 2+3=5. For ax² + bx + c with a≠1, use the AC method: find factors of ac that add to b, then split the middle term. Factoring is only possible over integers when D is a perfect square.
What is completing the square?+
Transform ax² + bx + c into a(x + h)² + k form. Steps: 1) divide by a; 2) move c to the right; 3) add (b/2a)² to both sides; 4) write the left side as a perfect square. This reveals the vertex (−h, k) of the parabola and derives the quadratic formula. Example: x²+4x+1=0 → (x+2)²=3 → x=−2±√3.
What are Vieta's formulas?+
For roots r₁ and r₂ of ax²+bx+c=0: r₁+r₂ = −b/a and r₁×r₂ = c/a. These hold without actually computing the roots. Example: x²−5x+6=0 → r₁+r₂=5, r₁×r₂=6, confirming roots 2 and 3. Vieta's formulas extend to higher-degree polynomials and are used in symmetric function theory and coding theory.
What does the vertex of a parabola represent?+
The vertex is the turning point: maximum if a < 0, minimum if a > 0. x-coordinate of vertex: −b/(2a). y-coordinate: c − b²/(4a), or substitute x back into the equation. In optimization problems, the vertex gives the optimal value — e.g., the price maximizing revenue, the angle maximizing projectile range.
How do complex roots arise in quadratic equations?+
When D = b²−4ac < 0, √D = i√|D| where i = √(−1). The roots are x = −b/(2a) ± i√(|D|)/(2a), a complex conjugate pair a ± bi. Complex roots always appear in conjugate pairs for real-coefficient polynomials. They represent the equation having no real x-intercepts but are essential in electrical engineering (impedance) and signal processing.
What is the difference between roots, solutions, and zeros?+
All three refer to values of x satisfying the equation: roots = solutions = zeros. "Zero of a function" means f(x) = 0; "root of an equation" means the equation equals zero; "solution" is the most general term. For the quadratic ax²+bx+c=0, roots are the x-values making the expression zero. The parabola y=ax²+bx+c crosses the x-axis at the real roots.
Can I solve a quadratic without the formula?+
Yes. Three alternatives: 1) Factoring (fastest when roots are simple integers). 2) Completing the square (gives vertex form too). 3) Graphing (approximate). 4) Numerically (Newton-Raphson iteration). The quadratic formula is the universal method — it always works and gives exact answers. For repeated use, the formula is most efficient.
What is a perfect square trinomial?+
A trinomial a²x²+2abx+b² that factors as (ax+b)². Recognizing it speeds up solving: x²+6x+9=(x+3)². The discriminant of a perfect square trinomial equals 0, confirming one repeated root. Example: 4x²−12x+9=(2x−3)², root x=3/2 (repeated). Completing the square transforms any quadratic into this form.
How are quadratics used in projectile motion?+
Height h(t) = h₀ + v₀t − ½gt², where g = 9.8 m/s². This is a quadratic in t. The vertex gives maximum height at t = v₀/g. Setting h(t) = 0 and solving the quadratic gives the time of flight. Example: ball thrown upward at 20 m/s from 2m height: 0 = 2 + 20t − 4.9t². D = 400 + 4(4.9)(2) = 439.2. Time of flight ≈ (−20 − √439.2)/(−9.8) ≈ 4.18 s.
What is the sum and product of roots formula for higher-degree polynomials?+
For the quadratic, Vieta's formulas are r₁+r₂=−b/a and r₁r₂=c/a. For a cubic x³+px²+qx+r=0 with roots r₁,r₂,r₃: sum=−p, sum of products in pairs=q, product=−r. Generally, the k-th elementary symmetric polynomial of n roots equals (−1)^k × (coefficient of x^(n−k)) / (leading coefficient). Used in coding theory, cryptography, and polynomial factorization.

References & Further Reading

  1. 1. Wolfram MathWorld — Quadratic Formula
  2. 2. Khan Academy — Quadratic Functions & Equations
  3. 3. Paul's Online Math Notes — Algebra
  4. 4. MIT OpenCourseWare — Mathematics
  5. 5. Al-Khwarizmi. Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala (c. 830 AD). Foundational algebra text.
  6. 6. Descartes, René. La Géométrie (1637). Introduced coordinate geometry linking algebra to parabolas.

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