T
Tools

Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula with step-by-step solutions

The calculator below solves the quadratic equation of

ax² + bx + c = 0

Modify the values and click the Calculate button to use

Fractional values such as 3/4 can be used.

Introduction to Quadratic Equations

In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:

ax² + bx + c = 0

where x is the unknown, a is the quadratic coefficient, b is the linear coefficient, and c is the constant. The numbers a, b, and c are known numbers, where a cannot be 0 (otherwise, the equation would be linear rather than quadratic).

There are various methods to solve quadratic equations, including factoring, using the quadratic formula, completing the square, and graphing. This calculator focuses on the quadratic formula and the basics of completing the square.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Derivation of the Quadratic Formula

Step 1: Start with the standard form

ax² + bx + c = 0

Step 2: Multiply by 1/a

x² + (b/a)x + (c/a) = 0

Step 3: Rearrange

x² + (b/a)x = -c/a

Completing the Square Relationship

From this point, it is possible to complete the square using the relationship that:

x² + bx + c = (x - h)² + k

Step 4: Complete the square

x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²

Simplify

Step 5: Simplify the right side

(x + b/2a)² = -4ac/4a² + b²/4a²

Step 6: Combine terms

(x + b/2a)² = (b² - 4ac) / 4a²

Step 7: Square root both sides

x + b/2a = ±√(b² - 4ac) / 2a

Step 8: Solve for x

x = (-b ± √(b² - 4ac)) / 2a

Explanation of the Formula

The ± sign in the quadratic formula indicates that there are two roots (solutions) to the equation. These x values are the points where the parabola (the graph of the quadratic equation) crosses the x-axis.

The quadratic formula also provides the axis of symmetry for the parabola, which is the vertical line that passes through the vertex. The axis of symmetry is given by:

x = -b / 2a

Real-World Applications:

  • Calculating areas: Finding dimensions of rectangles, circles, and other shapes
  • Projectile trajectories: Modeling the path of objects thrown or launched
  • Speed and distance: Solving problems involving motion and velocity
  • Optimization: Finding maximum or minimum values in various scenarios
  • Economics: Analyzing profit, cost, and revenue functions