Discover the definitive answer to, 'Does completing the square always work?' This powerful algebraic method is fundamental for solving quadratic equations. Our calculator demonstrates its universal application, allowing you to find both real and complex roots for any equation of the form ax² + bx + c = 0.
Formula:
A quadratic equation is typically written as ax2 + bx + c = 0.
Completing the square transforms this into the form (x + p)2 = q to facilitate solving for x.
The general steps are:
1. Divide the entire equation by 'a' (if a ≠ 1): x2 + (b/a)x + (c/a) = 0
2. Move the constant term to the right side: x2 + (b/a)x = -c/a
3. Add (b/(2a))2 to both sides to 'complete' the square on the left:
x2 + (b/a)x + (b/(2a))2 = -c/a + (b/(2a))2
4. Factor the left side into a perfect square and simplify the right side:
(x + b/(2a))2 = (b2 - 4ac) / (4a2)
5. Take the square root of both sides and solve for x:
x = (-b ± √(b2 - 4ac)) / (2a)
Where:
- a: The coefficient of the x2 term.
- b: The coefficient of the x term.
- c: The constant term.