📐 Completing the Square Calculator
Complete the square step by step
ax² + bx + c
How to Use This Calculator
Enter Coefficient a
Type the coefficient of x² (a). For example, if your equation is 2x² + 4x + 1, enter 2.
Enter Coefficient b
Type the coefficient of x (b). For 2x² + 4x + 1, enter 4. This can be zero if there's no x term.
Enter Constant c
Type the constant term (c). For 2x² + 4x + 1, enter 1. This can be zero if there's no constant.
Click Complete the Square
Press the button to convert the quadratic equation to vertex form and find the vertex.
View Vertex Form
See the equation in vertex form a(x-h)²+k and the vertex coordinates (h, k).
Formula
Standard Form: ax² + bx + c
Vertex Form: a(x - h)² + k
where h = -b/(2a) and k = c - (b²/(4a))
Example 1: Convert x² + 6x + 5 to vertex form
Given: a = 1, b = 6, c = 5
h = -b/(2a) = -6/(2×1) = -3
k = c - (b²/(4a)) = 5 - (36/(4×1)) = 5 - 9 = -4
Vertex Form: (x + 3)² - 4
Vertex: (-3, -4)
Example 2: Convert 2x² - 8x + 6 to vertex form
Given: a = 2, b = -8, c = 6
h = -(-8)/(2×2) = 8/4 = 2
k = 6 - (64/(4×2)) = 6 - 8 = -2
Vertex Form: 2(x - 2)² - 2
Vertex: (2, -2)
Example 3: Convert 3x² + 12x to vertex form
Given: a = 3, b = 12, c = 0
h = -12/(2×3) = -12/6 = -2
k = 0 - (144/(4×3)) = 0 - 12 = -12
Vertex Form: 3(x + 2)² - 12
Vertex: (-2, -12)
About Completing the Square Calculator
The Completing the Square Calculator converts quadratic equations from standard form (ax² + bx + c) to vertex form (a(x - h)² + k). This method is essential in algebra for solving quadratic equations, finding vertices, and understanding the properties of parabolas.
When to Use This Calculator
- Algebra Homework: Convert quadratic equations to vertex form
- Vertex Finding: Quickly find the vertex of a parabola
- Solving Quadratics: Use vertex form to solve equations
- Graph Analysis: Understand parabola transformations
- Optimization Problems: Find maximum or minimum values
- Exam Preparation: Practice completing the square method
Why Use Our Calculator?
- ✅ Shows Vertex: Displays vertex coordinates (h, k)
- ✅ Step-by-Step: Shows how h and k are calculated
- ✅ Vertex Form: Converts to a(x - h)² + k format
- ✅ 100% Accurate: Precise mathematical calculations
- ✅ Educational: Perfect for learning the method
- ✅ Completely Free: No registration required
Understanding Completing the Square
Completing the square is a technique that transforms a quadratic equation into a perfect square trinomial plus a constant. This reveals the vertex of the parabola and makes solving easier.
- The vertex (h, k) is the turning point of the parabola
- If a > 0, the vertex is the minimum point
- If a < 0, the vertex is the maximum point
- Vertex form makes graphing and transformations easier
- This method is used to derive the quadratic formula
Real-World Applications
Physics: Projectile motion equations use vertex form to find maximum height and time to reach it.
Business: Profit functions in vertex form show maximum profit and the production level that achieves it.
Engineering: Optimization problems often require completing the square to find optimal solutions.
Tips for Using This Calculator
- Coefficient a cannot be zero (then it's not quadratic)
- The vertex (h, k) shows the parabola's turning point
- Vertex form makes it easy to see translations and scaling
- Use this to verify hand-calculated vertex forms
- Remember: a(x - h)² means the parabola shifts h units horizontally
- The sign of a determines if vertex is max or min
Frequently Asked Questions
Why complete the square?
Completing the square converts equations to vertex form, making it easy to find the vertex, solve equations, and understand parabola transformations.
Can coefficient a be negative?
Yes! If a < 0, the parabola opens downward and the vertex is a maximum point instead of a minimum.
What if b = 0?
If b = 0, the equation is already in a form close to vertex form: ax² + c. The vertex is at (0, c).
How is this different from factoring?
Completing the square always works, even when equations don't factor easily. It also reveals the vertex, which factoring doesn't directly show.
What's the connection to the quadratic formula?
The quadratic formula is derived by completing the square on the general quadratic equation ax² + bx + c = 0.
Can I use this to solve equations?
Yes! Once in vertex form, you can set the equation equal to zero and solve for x. This is often easier than using the quadratic formula.
Is the vertex the same as the axis of symmetry?
The vertex lies on the axis of symmetry. The x-coordinate of the vertex (h) is the axis of symmetry line x = h.