Quadratic Equations

Calculator for solving quadratic equations using the quadratic formula - the core of algebra

Quadratic Equation Calculator

What is calculated?

This function solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Quadratic equations are degree 2 polynomials and describe parabolas.

Coefficients of the quadratic equation
ax² + bx + c = 0
a ≠ 0 required
Linear term
Absolute term
Solutions of the quadratic equation
Solution x₁
Solution x₂
Quadratic equations have 0, 1 or 2 real solutions

Quadratic Equations Info

Properties

Quadratic equations:

  • Degree 2 polynomial
  • At most 2 real solutions
  • Graphically: parabola
  • Quadratic formula applicable

Discriminant: The expression b² - 4ac determines the number of real solutions.

Solution types
D > 0: Two real solutions D = 0: One real solution D < 0: Two complex solutions
Examples
x² - 4 = 0:
x₁ = 2, x₂ = -2
x² - 2x + 1 = 0:
x = 1 (double root)
x² + 1 = 0:
Complex solutions: ±i
Related topics

→ Discriminant
→ Vertex form
→ Roots


Quadratic formula and variants

Standard form
\[ax^2 + bx + c = 0\] Quadratic equation
Quadratic formula
\[x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] General solution
Individual solutions
\[x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}\] \[x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}\]
Discriminant
\[D = b^2 - 4ac\] Determines solution type
Vertex form
\[f(x) = a(x - h)^2 + k\] Vertex: (h, k)
Factorized form
\[f(x) = a(x - x_1)(x - x_2)\] With roots x₁, x₂
Vieta's formulas
\[x_1 + x_2 = -\frac{b}{a}\] \[x_1 \cdot x_2 = \frac{c}{a}\]
Vertex
\[S = \left(-\frac{b}{2a}, -\frac{D}{4a}\right)\] Parabola vertex

Detailed worked example

Example: -2x² + 3x + 5 = 0

Given:

  • a = -2, b = 3, c = 5

Step 1 - Compute discriminant:

\[D = b^2 - 4ac = 3^2 - 4(-2)(5) = 9 + 40 = 49\]

Step 2 - Apply quadratic formula

\[x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{2(-2)} = \frac{-3 \pm 7}{-4}\]

Step 3a - First solution:

\[x_1 = \frac{-3 + 7}{-4} = \frac{4}{-4} = -1\]

Step 3b - Second solution:

\[x_2 = \frac{-3 - 7}{-4} = \frac{-10}{-4} = 2.5\]

Solutions: x₁ = -1, x₂ = 2.5

Check: -2(-1)² + 3(-1) + 5 = -2 - 3 + 5 = 0 ✓

Alternative solution methods

Different approaches to solve quadratic equations
Completing the square

Rewrite to vertex form

\[x^2 + px + q = 0\] \[\left(x + \frac{p}{2}\right)^2 = \frac{p^2}{4} - q\]
Factorization

Decompose into linear factors

\[ax^2 + bx + c = a(x - x_1)(x - x_2)\]
p-q formula

For normalized form x² + px + q = 0

\[x_{1,2} = -\frac{p}{2} \pm \sqrt{\frac{p^2}{4} - q}\]
Graphical solution

Read off roots of the parabola

Intersections with the x-axis

Geometric interpretation

Parabolas and their properties
D > 0: Two roots

Parabola intersects x-axis twice

D = 0: One root

Parabola is tangent to the x-axis

D < 0: No real roots

Parabola lies above the x-axis

Parabola properties

Opening:

  • a > 0: Parabola opens upward
  • a < 0: Parabola opens downward
  • |a| large: narrow parabola
  • |a| small: wide parabola

Special points:

  • Vertex: (-b/2a, -D/4a)
  • y-intercept: (0, c)
  • Axis of symmetry: x = -b/2a
  • Roots: quadratic formula

Practical applications

Physics
  • Projectile motion
  • Free fall
  • Oscillations
  • Optical devices
Economics
  • Profit functions
  • Cost optimization
  • Demand curves
  • Break-even analysis
Engineering
  • Bridge construction
  • Antenna engineering
  • Computer graphics
  • Control engineering

Historical context

Development of solution formulas
  • Babylonians (~2000 BC): Geometric methods
  • Al-Khwarizmi (~830): Systematic algebra
  • Bombelli (1572): Complex numbers
  • Quadratic formula: Modern standard form
Modern significance
  • Optimization: Extremum problems
  • Modeling: Parabolic processes
  • Computers: Numerical methods
  • AI/ML: Loss functions
Summary

Quadratic equations and the quadratic formula are central concepts in algebra with enormous practical importance. From ancient geometry to modern applications in AI and engineering they bridge theoretical mathematics and real-world problem solving.




Mathematical Equations

Associative Property  •  Linear equation  •  Quadratic equation  •  Cubic equation  •  Discriminant  •  Gauss elimination method  •