Convert quadratic equations to vertex form and find solutions step-by-step.
Last updated: 2026-05-24T22:58:31.500Z
Enter coefficients a, b, c as numbers; a must be non-zero.
Vertex Form
(x + 3)²
+ 4
Solutions
x₁ = -1
x₂ = -5
Completing the square is a technique to solve quadratic equations and convert them to vertex form. The method involves adding a specific value to both sides of the equation to create a perfect square trinomial, which factors as (x + p)². This reveals the parabola's structure directly.
This technique is fundamental in algebra and calculus. It's used to derive the quadratic formula, find parabola vertices, integrate functions, and solve optimization problems. The vertex form a(x - h)² + k immediately reveals the vertex at (h, k) and the axis of symmetry at x = h.
Solve: x² + 6x + 5 = 0
Step 1: Move constant: x² + 6x = -5
Step 2: Half of 6: 6 ÷ 2 = 3
Step 3: Square it: 3² = 9
Step 4: Add to both sides: x² + 6x + 9 = -5 + 9 = 4
Step 5: Factor: (x + 3)² = 4
Step 6: Take √: x + 3 = ±2
Step 7: Solve:
x + 3 = 2 → x = -1
x + 3 = -2 → x = -5
Vertex form: (x + 3)² - 4, Vertex: (-3, -4)
Use it to find the vertex, convert to vertex form, solve when factoring is difficult, or derive the quadratic formula. It's essential for optimization problems.
First divide the entire equation by 'a' to make the x² coefficient equal to 1. This creates a standard form for completing the square.
This creates a perfect square trinomial on the left. The pattern x² + bx + (b/2)² factors as (x + b/2)², which is the key to the technique.
The quadratic formula is derived by completing the square on ax² + bx + c = 0. That's why completing the square is so fundamental.
When the discriminant (b² - 4ac) is negative, the square root of a negative number appears, giving complex solutions in the form p ± qi.
Yes! Every quadratic equation can be solved by completing the square. It always works, making it a universal technique.
Vertex form a(x-h)² + k immediately shows the vertex (h,k), minimum/maximum value (k), axis of symmetry (x=h), and direction of opening.
Vertex form makes graphing easy. You get the vertex, then plot points symmetric about x = h. The x² coefficient determines width and direction.
Related Tools
Solve absolute value equations.
Solve absolute value inequalities.
Apply associative property.
Multiply using box method.
Find complex roots.
Solve using Cramer's Rule.