Complete the square Solver - Free Equation Calculator

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

Solving quadratic equations will never be the same again. A quadratic equation of the form >ax^2+ bx + c = 0 for x \ne 0 can now be solved online using this equation calculator. The calculator is free, accurate and efficient. The Best part is that the calculator shows all the steps alongside explanation of how to arrive at the solution. A quadratic equation is said to be a square if it can be reduced into a perfect square of first degree polynomials. For example the equation

X^-2x+1=0 can be expressed as (x-1)^2=0. Note that (x-1)^2 = X^-2x+1

From the example above, it is easy to find the roots of a square function. The process is as simple as taking the square root of either side of the equation.

In mathematics, nice quadratics (those that are perfect squares) are a rare occurrence. However, it is possible to create a perfect square out of any equation simply by adding a constant.

How to complete the square with a coefficient

You can create a perfect square just by adding a constant on the LHS of the equation. To balance the action, you have to subtract similar value on the LHS.

First, you have to determine the right coefficient, one that is capable of delivering a perfect square. Let’s illustrate this idea using an example. Considering our famous example ax² + bx + c = 0, just by adding and subtracting the square of \frac{b}{2}, it is sufficient to form a perfect square, assuming a=1.

Lets illustrate this further using a live example

Example: 2x^2-6x-5=0

Step 1: We write the polynomial/equation into standard form with leading coefficient equal 1 (Divide all coefficients by 2​):

x^2-3x-\frac{5}{2}=0

Step 2: Introduce a new Constant: -\frac{3}{2} ​​ on the LHS to form a Perfect Square

(x^2-3x+(-\frac{3}{2})^2)-\frac{5}{2}-(-\frac{3}{2})^2=0

Step 3: Simplify the expression

(x-\frac{3}{2})^2-\frac{19}{4}=0

Step 4: Take Square root on both Sides of the Equation & Simplify

x-\frac{3}{2}=\pm \sqrt{\frac{19}{4}}

Step 5: Solution

x=\frac{(3+\sqrt{19})}{2}   OR  x=\frac{(3-\sqrt{19})}{2}

Notice that in the example above we first divide through by 2 to reduce the expression into a standard form equation. With a standard equation, it is now possible to apply the completing square procedure to determine the solution.

Completing square formula Examples with Solution

Completing the square with fractions

As I mentioned above, nice quadratics are rare occurrence. The first step into completing square method is to divide through by the coefficient of x^2. If the other coefficients are not divisible by a, then the result is a fraction. A completing square formula calculator that works with fractions is now available online. The calculator shows you all the steps, irrespective of the nature of the coefficients.