Quadratic Equation Solver

Solve ax² + bx + c = 0

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The term "quadratic" comes from "quadratus," the Latin word for square, because the variable is squared.

The solutions to a quadratic equation are called roots, and they represent the x-values where the parabola (the graph of the quadratic function) crosses the x-axis. A quadratic equation can have two real roots, one repeated real root, or two complex conjugate roots.

How to Calculate - Algebra Guide #2 - Quadratic Equations

Follow these detailed steps:

  1. Step 1: Identify Coefficients
    Write equation in standard form ax² + bx + c = 0 and identify a, b, c. For 2x² - 8x + 6 = 0: a=2, b=-8, c=6.
  2. Step 2: Calculate the Discriminant
    Find b² - 4ac. This determines the nature of solutions: positive = 2 real roots, zero = 1 repeated root, negative = complex roots.
  3. Step 3: Apply the Quadratic Formula
    Use x = (-b ± √(b²-4ac)) / 2a. For our example: x = (8 ± √(64-48)) / 4 = (8 ± 4) / 4, giving x = 3 or x = 1.

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

The discriminant (b² - 4ac) determines the nature of the roots:

  • If b² - 4ac > 0: Two distinct real roots
  • If b² - 4ac = 0: One repeated real root
  • If b² - 4ac < 0: Two complex conjugate roots

Example

Standard Quadratic Example

Problem: Solve x² - 5x + 6 = 0

Solution:

  1. Identify: a = 1, b = -5, c = 6
  2. Discriminant: (-5)² - 4(1)(6) = 25 - 24 = 1
  3. x = (5 ± √1) / 2 = (5 ± 1) / 2
  4. x₁ = (5 + 1) / 2 = 3
  5. x₂ = (5 - 1) / 2 = 2
  6. Solutions: x = 3 or x = 2

Why This Calculation Matters

Quadratic equations appear everywhere in physics, engineering, and real-world problem solving. From calculating projectile motion to optimizing areas, the quadratic formula is your universal tool for solving any quadratic equation.

Real-World Application Scenarios

Algebra Guide #2 - Quadratic Equations - Here are practical situations where you'll use this calculation:

  • Projectile Motion: A ball thrown upward from 6ft with velocity 48 ft/s has height h = -16t² + 48t + 6. When does it hit ground? Solve -16t² + 48t + 6 = 0.
  • Area Optimization: A rectangle has perimeter 20 and width x. Area = x(10-x) = 10x - x². Maximum area when derivative = 0, or vertex at x = 5.
  • Profit Maximization: Revenue = price × quantity. If demand = 100 - 2p and supply creates R = p(100-2p) = 100p - 2p², find optimal price.
  • Engineering Design: A bridge arch has equation y = -0.01x² + 0.4x. Finding where y = 0 gives the span width of the arch.

Quick Calculation Tips

  • Always check the discriminant first to know what type of solutions to expect
  • If b² - 4ac is a perfect square, the roots are rational and might factor easily
  • The vertex form y = a(x-h)² + k gives you the maximum/minimum point directly
  • Graphing calculators can verify your algebraic solutions visually

Common Mistakes to Avoid

  • Forgetting the ± sign
    Quadratic equations typically have TWO solutions. The ± means you get both roots.
  • Sign errors in the discriminant
    Pay careful attention to negative b values: if b = -8, then b² = 64 (positive).

Frequently Asked Questions

What is the discriminant?

The discriminant is the expression b² - 4ac under the square root in the quadratic formula. It tells us how many and what type of roots the equation has.

What if a = 0?

If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Use our Solve for X calculator for linear equations.

Can quadratic equations have complex roots?

Yes! When the discriminant is negative, the roots are complex numbers involving the imaginary unit i. For example, x² + 1 = 0 has roots x = ±i.

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