Quadratic Equations

Executive Summary

Quadratic equations are second-degree polynomials of the form ax² + bx + c = 0. Solving them determines the roots (x₁ and x₂) and defines the shape and vertex of the corresponding parabola.

Key Takeaways
  • root solving: Quadratic equations are ax² + bx + c = 0 solved using the quadratic formula.
  • discriminant check: The discriminant b² - 4ac determines the number and type of roots (real or complex).
  • parabolic shapes: Real roots represent the x-intercepts where the parabola crosses the horizontal axis.
Formula
x = [ -b ± √(b² - 4ac) ] / 2a

Quadratic Roots Formula

Variable Glossary
SymbolMeaning & Description
xRoots of the equation
a, b, cPolynomial coefficients
DDiscriminant = b² - 4ac
Step-by-Step Worked Example
1. Mapped Variables
Coefficient a
a1
Coefficient b
b-5
Coefficient c
c6
2. Equation Substitution
Equation with standard inputs
x = [ -(-5) ± √((-5)² - 4·(1)·(6)) ] / (2·1)
3. Calculation Steps
Step 1: Calculate Discriminant (D)

D = b² - 4ac = (-5)² - 4·(1)·(6) = 25 - 24 = 1

Step 2: Evaluate square root of Discriminant

√D = √1 = 1.0000

Step 3: Solve roots

Roots = [ -b ± √D ] / 2a = [ 5 ± 1.0000 ] / 2 = x = 3.0000, x = 2.0000

Final Resolved Equation Rootsx = 3.0000, x = 2.0000
Live Simulation

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Coefficient a1
Coefficient b-5
Coefficient c6
Calculating...
Common Mistakes to Avoid

Sign Errors in Substitution: Negative coefficients (especially for 'b' and 'c') are common sources of errors. E.g., when subtracting 4ac, if c is negative, the double negative turns the operation into addition.

Frequently Asked Questions
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