Solving By Completing The Square
Solve quadratic equations ax^2+bx+c=0 by completing the square. Get step-by-step solutions, discriminant, and vertex form a(x-h)^2+k with this free calculator.
About This Calculator
This solving by completing the square calculator helps you solve quadratic equations of the form ax^2 + bx + c = 0 using the completing the square method. Whether you are a high school student learning algebra, a college student preparing for exams, or a teacher demonstrating quadratic solutions, this tool provides instant results with a full step-by-step breakdown.
The completing the square method works by rewriting the quadratic equation so that part of it forms a perfect square trinomial, which can then be factored as (x + p)^2. This method works for every quadratic equation -- including those with irrational roots or complex numbers -- making it more powerful than basic factoring. The calculator converts your equation into the vertex form a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola, giving you deeper insight into the equation's graph and properties.
How the Calculation Works
Given the quadratic equation ax^2 + bx + c = 0, the calculator follows these steps:
- Move the constant term c to the right-hand side: ax^2 + bx = -c
- Divide both sides by a (if a ≠ 1) to normalize the x^2 coefficient
- Add (b/(2a))^2 to both sides to create a perfect square trinomial on the left
- Rewrite the left side as (x + b/(2a))^2
- Take the square root of both sides and solve for x
The discriminant D = b^2 - 4ac determines the nature of the solutions: two distinct real roots (D > 0), one repeated real root (D = 0), or two complex conjugate roots (D < 0).
Regional Notes
Completing the square is taught universally in secondary school algebra curricula. In India (CBSE/ICSE), it is covered in classes 9-10 as part of quadratic equations. In the US, it appears in Algebra I and II courses (typically grades 8-11). In the UK, it is part of GCSE and A-Level mathematics. The method is identical across all regions -- only the curriculum timing differs.
Frequently Asked Questions
What is completing the square in algebra?
Completing the square is a method used to solve quadratic equations by rewriting ax^2+bx+c=0 in the form a(x-h)^2+k=0. This transformation makes it easier to solve for x by isolating the squared term and taking the square root of both sides.
How do you solve a quadratic equation by completing the square?
To solve a quadratic equation by completing the square, first move the constant term to the right side. Then add (b/2a)^2 to both sides so the left side becomes a perfect square trinomial. Rewrite it as (x + b/2a)^2, take the square root of both sides, and solve for x.
What if the coefficient a is not equal to 1?
If the coefficient a is not 1, divide the entire equation by a first. This normalizes the x^2 coefficient to 1, allowing you to proceed with the standard completing the square steps on the simplified equation x^2 + (b/a)x + c/a = 0.
Can completing the square solve equations with complex roots?
Yes, when the discriminant b^2-4ac is negative, the equation has two complex conjugate roots. The completing the square method still works by using imaginary numbers: x = -b/2a +/- i·sqrt(|D|)/2a, where i is the imaginary unit.
How is the discriminant related to completing the square?
The discriminant D = b^2-4ac determines the nature of the roots. If D > 0, there are two distinct real solutions. If D = 0, there is one repeated real solution (double root). If D < 0, there are two complex solutions. The discriminant appears naturally when taking the square root in the final step of completing the square.
What is the completed square form a(x-h)^2+k?
The completed square form a(x-h)^2+k expresses a quadratic equation where (h,k) is the vertex of the parabola. Here h = -b/2a and k = c - b^2/4a. This form is useful for graphing quadratics, finding the vertex, and determining the axis of symmetry.
Is the solving by completing the square calculator free to use?
Yes, this calculator is completely free to use with no registration or hidden costs. It works for any quadratic equation with real or complex roots. Simply enter the coefficients a, b, and c, and get instant step-by-step solutions.
What is the difference between factoring and completing the square?
Factoring works only when a quadratic can be expressed as a product of linear factors (when discriminant is a perfect square). Completing the square works for all quadratic equations, including those with irrational or complex roots. It also directly reveals the vertex of the parabola, making it more versatile than basic factoring.