Completing the Square Calculator
Solve quadratic equations of the form ax² + bx + c = 0 with detailed steps
Tip: Enter coefficients below and get the solution with all algebraic steps shown.
Enter Quadratic Equation Coefficients
Step-by-Step Solution
Step 1: Normalize the Equation
Step 2: Move Constant Term
Step 3: Complete the Square
Step 4: Factor Perfect Square
Step 5: Solve for x
Final Solutions
These are the roots of the quadratic equation.
Understanding Completing the Square
Why Use This Method?
- Reveals the vertex form of the parabola
- Works when factoring is difficult
- Foundation for the quadratic formula
- Essential for conic sections in geometry
Key Concepts
- (b/2)² determines the completing value
- Creates a perfect square trinomial
- Vertex form: a(x-h)² + k
- Reveals maximum/minimum points
"Completing the square is not just an algebraic technique - it's a fundamental concept that connects algebra to geometry through the study of parabolas and other conic sections."
Applications of Completing the Square
Quadratic Functions
Converting to vertex form makes graphing parabolas straightforward. The vertex (h,k) becomes immediately visible in the equation y = a(x-h)² + k, showing the maximum or minimum point of the parabola.
Physics Problems
Many projectile motion equations are quadratic in nature. Completing the square helps determine maximum height, time of flight, and range of projectiles in physics applications.
Optimization
Business and economics problems often require finding maximum profit or minimum cost points. These optimization problems frequently involve quadratic models solved by completing the square.
Did You Know? The quadratic formula (x = [-b ± √(b²-4ac)]/2a) is derived by completing the square on the general quadratic equation ax² + bx + c = 0.
Frequently Asked Questions
What if my 'a' coefficient isn't 1?
Our calculator handles any 'a' value automatically. The first step divides all terms by 'a' to normalize the equation. This is shown clearly in the step-by-step solution.
How do imaginary solutions appear?
If the discriminant (b²-4ac) is negative, you'll get complex solutions involving 'i'. Our calculator displays these properly, like "2 ± 3i" for non-real roots.
Can I see decimal approximations?
The calculator shows exact form solutions by default (with square roots). For decimal approximations, simply evaluate the square roots using a basic calculator.
Why does completing the square work?
The method transforms the equation into a perfect square trinomial, which can be written as (x + d)². This geometric technique dates back to ancient mathematicians.
When should I use this vs. the quadratic formula?
Use completing the square when you need vertex form or exact solutions. The quadratic formula is faster for just finding roots, but doesn't show the algebraic process.