Finding the Vertex
Vertex form \(a(x-h)^2 + k\) directly gives vertex \((h, k)\).
Completing the Square
Transform quadratic equations to vertex form step-by-step
Master the technique of completing the square to rewrite quadratics in vertex form \(a(x - h)^2 + k\).
Completing the Square Calculator
Enter a quadratic equation in standard form \(ax^2 + bx + c\) to transform it to vertex form.
Original Equation:
\[x^2 + 6x + 5 = 0\]
Vertex Form Result:
Vertex Form:
\[(x + 3)^2 - 4 = 0\]
Vertex (h, k):
\[(-3, -4)\]
Solutions:
\[x = -1, x = -5\]
Axis of Symmetry:
\[x = -3\]
How to Complete the Square
For \(ax^2 + bx + c = 0\):
- Divide by \(a\) if \(a \neq 1\)
- Move constant term: \(x^2 + \frac{b}{a}x = -\frac{c}{a}\)
- Add \((\frac{b}{2a})^2\) to both sides
- Factor the left side as a perfect square
- Take square root and solve
Key Insight
Completing the square derives the quadratic formula!
Applications
Graphing Parabolas
Vertex form makes graphing easy - start at vertex, use symmetry.
Optimization
Find maximum/minimum values in real-world problems.
Worked Examples
\(x^2 + 8x + 7 = 0\)
Step 1: Move constant: \(x^2 + 8x = -7\)
Step 2: Half of 8 is 4, square: \(4^2 = 16\)
Step 3: Add: \(x^2 + 8x + 16 = 9\)
Step 4: Factor: \((x+4)^2 = 9\)
Solutions: \(x = -1, x = -7\)
\(2x^2 - 12x + 10 = 0\)
Step 1: Divide by 2: \(x^2 - 6x + 5 = 0\)
Step 2: Move constant: \(x^2 - 6x = -5\)
Step 3: Add 9: \((x-3)^2 = 4\)
Solutions: \(x = 1, x = 5\)
Frequently Asked Questions
Why complete the square?
It derives the quadratic formula, gives vertex form directly, and helps understand parabola geometry.
How is it related to the vertex?
The vertex form comes directly from completing the square: \(h = -b/(2a)\), \(k = c - b^2/(4a)\).
When is it required?
Essential for deriving the quadratic formula, converting forms, calculus integrals, and finding circle centers.