Completing the square formula calculator

A Worked example to illustrate how the complete square calculator Works:

Completing the square formula calculator

A quadratic equation of the form ax² + bx + c = 0 for x, where a \ne 0 can be solved using the completing the square formula with this calculator. It is an online calculator capable of completing the square of a quadratic equation with all the steps.

Naturally some quadratics can be factored into a perfect square. For example

X^2-2x+1=0 can be factored into its constituent factor polynomials of degree 1 as follows:

(x-1)^2=0

With the later form, it is easy to solve the equation simply by taking square roots on either side of the expression. Doing so, reduces the LHS into a first degree polynomial which is solvable through factorization and simplification.

Nice quadratics (factorable equations) are not always the case. In fact, most quadratics cannot be factored into first degree polynomials. Nevertheless, we can force perfect squares into such equations simply by adding a constant.

For Example x^2+6x+7=0 is not a perfect square. Nevertheless, we can create a perfect square out of it simply by adding 2. The result will be (x+3)^2=2 which is easier to solve than the earlier form.

Discriminant and completing Square

The discriminant of a quadratic is a constant D= (b^2-4ac). The constant is useful in determining the nature of the solution to a quadratic even before we engage into solving it. If a quadratic equation has a positive discriminant, then it has two roots. If D=0, then there is a possibility of one non-zero solution. Lastly, if D is negative, the equation will have complex roots. A complex root will have both the real and the imaginary part.

Quadratic formula calculator imaginary

A quadratic equation that has a negative discriminant will always have at least 1 complex root. This calculator will help you to solve quadratic equations using the complete square formula even when the roots are complex numbers.

Here are some more examples to illustrate how the calculator works.

Examples above have been worked using EquationCalc. The calculator gives all the steps alongside the necessary explanation on how we arrive at an answer. You can learn algebra online using this equation calculator. To practice, simply solve your math before confirming the solution using the calculator.