How to Complete the Square

Completing the square is an important skill in maths and you may find yourself frequently having to solve problems that involve completing the square, particularly in algebra. This guide will show you how to complete the square!

What is Completing the Square?

Completing the square is an algebraic process that allows you to convert a quadratic function in standard form to vertex form.  Completing the square involves adding a value to and subtracting a value from a quadratic polynomial so that it contains a perfect square trinomial. You can then rewrite this trinomial as the square of a binomial.

Definition of Polynomial and Perfect Square trinomial

So lets firstly clear up some terms.

What is a polynomial?

A polynomial is an expression of more than two algebraic terms, especially the  sum of several terms that contain different powers of the same variable(s).

For example:

5xy + 2x - 7

5xy is the 1st term, 2x is the 2nd term and -7 is the third term.

A polynomial can have constants ( like the numbers 5 and -11), variables (like x and y), exponents that are positive whole numbers, (like the 2 in x²). But a polynomial cannot have division by a variable, so an expression with 2/(x+2) is not a polynomial.

A trinomial by our definition of a polynomial therefore is an algebraic expression consisting of three terms.

A perfect square trinomial looks like this:

x² + 14x+49

The above trinomial is the algebraic expression we get when we expand

(x+7)² or (x+7)(x+7).

So to make a perfect square trinomial, we square the first term ( of the binomial) (x+7)².

Then we twice the product of the binomials first and last terms. Finally we square the last term of the binomial.

so we get:

x² + 14x+49

How to Complete the Square - Step By Step

So back to how to complete the square. Let us complete the square of:

y= x² + 6x +5  (this function is in standard form.)

For the function y= x² + 6x +5 to complete the square:

Group the First two terms.

y= (x² + 6x)

Inside the brackets, add and subtract the square of half the coefficient of the  second term.

y= (x² + 6x +9 -9)

Group the perfect square trinomial.

y= (x² + 6x +9) -9 + 5


Reunite the perfect square trinomial as the square of a binomial.

y= (x+3)² -9 + 5

Simplify.

y= (x+3)² -4  (This function is in vertex form.)

Remember:

y= x² +6x +5  and   y= (x+3)² -4 represent the same quadratic function. You can use both forms of the function to determine what the graph will look like. However an added benefit of the vertex form is of course that you can identify the vertex of the parabola without graphing.


How to Complete the Square with fractions

  • y= 4x² -28x -23
  • y= 4(x² -7x) -23
  • y= 4[(x² -7x + (7/2)² -(7/2)²]  -23
  • y= 4(x² -7x + 49/4 -49/4)  -23
  • y= 4[(x² -7x + 49/4)-49/4]  -23
  • y= 4[(x-7/2)²-49/4]  -23
  • y= 4(x-7/2)²-4(49/4)  -23
  • y= 4(x-7/2)²-49  -23
  • y= 4(x-7/2)²-72

Completing the square can be used with fractions (as shown above) and also decimals.

So that was my guide on completing the square. Be sure to review these steps as often as you need to sharpen up your skills on this algebraic process.

Thanks for joining me!

Roli Edema.



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