Completing the Square Calculator for Quadratics
Enter the coefficients of a quadratic equation to solve it by completing the square step by step
| Property | Value |
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Step by Step Solution
Completing the square calculator solves quadratics step by step
Completing the square calculator tools let you solve any quadratic equation in the form a x squared plus b x plus c equals zero by transforming it into a perfect square binomial. This calculator shows every algebraic step and reports real or complex roots depending on the equation.
How the completing the square calculator works
The completing the square calculator works by first dividing all terms by a if a is not equal to 1, then moving the constant term to the right side, adding the square of half the b coefficient to both sides, and rewriting the left side as a squared binomial before solving for x.
Completing the square formula explains the algebra
Completing the square formula explains how a quadratic expression can be rewritten as a perfect square plus or minus a constant. For an equation in the form x squared plus bx plus c equals zero, the formula adds the square of b over 2 to both sides to create a perfect square trinomial.
The general formula
Solving for x
When a is not equal to 1
If the leading coefficient a is not 1, divide every term of the equation by a first. For example, 2x squared minus 12x plus 7 equals 0 becomes x squared minus 6x plus 3.5 equals 0 after dividing by 2, and the standard steps can then be applied.
Reference table shows example quadratic solutions
Reference table data below shows solutions for several example quadratic equations solved by completing the square.
| Equation | x1 | x2 |
|---|---|---|
| x² – 4 = 0 | 2 | -2 |
| x² – 6x + 3.5 = 0 | 5.4495 | 0.5505 |
| x² + 2x + 5 = 0 | -1 + 2i | -1 – 2i |
| 2x² – 12x + 7 = 0 | 5.4495 | 0.5505 |
| x² – 5x + 6 = 0 | 3 | 2 |
Real world quadratic equations apply to motion and design
Real world quadratic equations come up in physics problems involving projectile motion, in engineering problems involving area and optimization, and in finance problems involving growth rates. Completing the square provides a reliable method to solve these equations even when factoring is not possible.
Common uses for completing the square
- Solving quadratic equations that do not factor with whole numbers
- Deriving the vertex form of a parabola for graphing
- Finding maximum or minimum values in optimization problems
- Working with equations that have complex or irrational roots
Frequently asked questions about completing the square
Completing the square is a method for solving quadratic equations by rewriting the equation so that one side becomes a perfect square binomial. This makes it possible to solve for the variable by taking the square root of both sides.
For an equation in the form x squared plus bx plus c equals zero, completing the square involves adding and subtracting the quantity b divided by 2, squared. This transforms the equation into the form open parenthesis x plus b over 2 close parenthesis squared equals b over 2 squared minus c.
When a is not equal to 1, divide every term in the equation by a before completing the square. This converts the equation into the standard form x squared plus bx plus c equals zero so the usual steps can be applied.
Yes if the value under the square root becomes negative during the process, the solutions are complex numbers involving the imaginary unit i. This happens when the discriminant of the quadratic equation is negative.
Completing the square is the method used to derive the quadratic formula, so both approaches always produce the same solutions. Completing the square shows each algebraic step individually, while the quadratic formula applies the final derived result directly.
Completing the square should be used when a quadratic equation cannot be factored easily using whole numbers. It works for every quadratic equation, including those with irrational or complex solutions, where simple factoring methods fail.