Quadratic Equation Solver

Solve any quadratic equation of the form ax² + bx + c = 0. Get step-by-step solutions, graphical representation, and detailed explanations.

Enter Coefficients

Must not be zero

Recent Equations

No equations saved yet.

Solution

Equation:
1x² -3x +2 = 0
Discriminant (D):
D = b² - 4ac = 1
Two distinct real roots
Solutions:
x₁ = 2.0000
x₂ = 1.0000

Key Properties

Vertex: (1.5000, -0.2500)
Axis of Symmetry: x = 1.5000
Y-Intercept: (0, 2)
Direction: Opens upward

Step-by-Step Solution

Step 1: Given quadratic equation:
x² -3x +2 = 0
Step 2: Calculate the discriminant (D = b² - 4ac):
D = (-3)² - 4 × 1 × 2
D = 1
Step 3: Determine the nature of roots:
Two distinct real roots
Step 4: Apply the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a
x = [3 ± √1] / 2
Step 5: Calculate the two solutions:
x₁ = [3 + 1.0000] / 2 = 2.0000 x₂ = [3 - 1.0000] / 2 = 1.0000
x₁ = 2.0000, x₂ = 1.0000
Step 6: Find the vertex of the parabola:
x = -b/(2a) = --3/(2×1) = 1.5000 y = f(1.5000) = 1×(1.5000)² + -3×1.5000 + 2 = -0.2500
Vertex: (1.5000, -0.2500)
Step 7: Find the axis of symmetry:
x = -b/(2a) = 1.5000
Axis of symmetry: x = 1.5000
Step 8: Find the y-intercept:
Set x=0: y = 1×(0)² + -3×0 + 2 = 2
y-intercept: (0, 2)
Back to Home

Explore More Math Tools

Discover other useful calculators in our collection:

LCM & GCF Calculator

Find least common multiple and greatest common factor

Percentage Calculator

Calculate percentages, increases, and decreases

Unit Converter

Convert between different measurement units

Math Puzzle Generator

Challenge yourself with math puzzles

Sudoku Generator

Generate and solve Sudoku puzzles

Arithmetic Speed Test

Test and improve your mental math skills

Mastering Quadratic Equations

Quadratic equations are fundamental in algebra and appear in many advanced mathematical concepts. Understanding how to solve them is essential for students and professionals in STEM fields.

Why Quadratic Equations Matter

Foundation for Advanced Math

Quadratic equations serve as a bridge between basic algebra and more complex mathematical concepts like calculus, linear algebra, and differential equations.

Real-World Applications

From physics to finance, quadratic equations model many real-world phenomena including projectile motion, optimization problems, and economic forecasting.

Critical Thinking Development

Solving quadratic equations requires logical reasoning and problem-solving skills that are valuable in many areas beyond mathematics.

Standardized Test Preparation

Quadratic equations are a key component of many standardized tests including the SAT, ACT, GRE, and GMAT, making mastery essential for academic success.

Tips for Solving Quadratic Equations

  • Always check if the equation can be factored before using the quadratic formula
  • Remember that if a = 0, the equation is linear, not quadratic
  • Pay attention to the discriminant to understand the nature of the roots
  • When graphing, the vertex represents either the maximum or minimum value of the quadratic function
  • Practice recognizing perfect square trinomials to simplify solving by completing the square
  • Double-check your solutions by plugging them back into the original equation

Common Mistakes to Avoid

  • Forgetting that the quadratic formula requires the equation to be in standard form (ax² + bx + c = 0)
  • Mishandling negative signs, especially when b or c is negative
  • Incorrectly calculating the discriminant (b² - 4ac)
  • Not simplifying radicals completely when possible
  • Forgetting to consider both the positive and negative square root in the quadratic formula
  • Dividing incorrectly when simplifying the final answer