Completing Squares Calculator


With this calculator, you can complete the square of an algebraic expression. By entering the algebraic expression in the input box, the calculator will return a version with the complete square if possible.

How to use the calculator to complete the square?

Step 1: The algebraic expression must be entered in the corresponding input box. You must use * to indicate multiplication between variables and coefficients. For example, instead of 5x or 2x^2, enter 5*x or 2*x^2.

Step 2: Click “Complete” to get a simplified version by completing the square of the entered expression.

Step 3: The solution along with the algebraic expression entered will be displayed at the bottom. If it is not possible to complete the square of the entered algebraic expression, the expression will be displayed in its original form or in a simplified form if possible.

How to enter expressions in the calculator?

To enter expressions, you must use the * sign to indicate multiplication between coefficients and variables. Additionally, use ^ to indicate an exponent. For example,

  • If you want to write \(x^2+4x+2\), enter x^2+4*x+2.
  • If you want to write \(3x^2-9x+11\), enter 3*x^2-9*x+11.
  • If you want to write \(\frac{1}{3}x^2-\frac{1}{2}x+2\), enter 1/3*x^2-1/2*x+2.

Notice that we can also use fractional coefficients. Simply enter the fraction in the form 1/2. This must be followed by the variable. For example, 1/2*x indicates a half of x.

What does it mean to complete the square?

Completing the square is the process of taking a quadratic equation from the form \( ax^2+bx+c=0\) to the form \( a(x+d)^2+e=0\).

How to complete the square?

If we have a quadratic expression \( 2x^2+12x+6=0\), we can complete the square by following these steps:

  1. We rearrange the expression to move all constant terms to the left side. Then, we would have \( 2x^2+12x=-6\).
  2. Now, we divide the entire expression by the coefficient a. Therefore, we have \( x^2+6x=-3\).
  3. Then, we add\((\frac{b}{2})^2\) to both sides of the expression. In this case, \((\frac{6}{2})^2=9\). Therefore, we have \( x^2+6x+9=-3+9\).
  4. Now, we can factor the expression on the left-hand side and simplify the expression on the right-hand side. Doing this, we have \( (x+3)^2=6\).

In case we needed to solve the equation for x, we would simply have to take the square root of both sides of the equals sign. We would obtain a linear equation that can be easily solved.

What is the purpose of completing squares?

Completing squares can be helpful when solving quadratic equations. We can isolate the squared expression and then take the square root of both sides of the equation to find the solution. We can see this in the example shown above, where taking the square root would give us a linear equation that is easy to solve.

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