Up

Completing the square by finding the constant

You remember that a perfect square trinomial is one that can be factored into the square of a binomial.  For example, the trinomial \({x^2} + 4x + 4\) is a perfect square trinomial because

\({x^2} + 4x + 4 = {\left( {x + 2} \right)^2}\)

Sometimes it is convenient when solving problems to be able to work with perfect square trinomials.  But not all trinomials are perfect squares.  Thankfully, a technique exists to overcome this situation.  It is called completing the square.  Here’s how it works.

For the polynomial \({x^2} + bx + c\) (note here that the coefficient on \({x^2}\) is 1)we can complete the square by

  • Find the value of \(\Large \frac{b}{2}\)
  • Square this value to obtain \({\left( {\Large \frac{b}{2}} \right)^2}\)
  • Set this value as c.
  • You will obtain a perfect square trinomial that can be factored as \({\left( {x + \left( {\Large \frac{b}{2}} \right)} \right)^2}\)

Let’s take a look at an example.

Example:  Find the value of c that completes the square for \({r^2} + 11r + c\).

Solution:  Here the coefficient on r (that is, the value b) is 11.  Following the steps for completing the square we find

  • The value of \(\Large \frac{b}{2}\) is \(\Large \frac{{11}}{2}\)
  • Squaring this value, we obtain \(\Large \frac{{121}}{4}\)
  • So \(c = \Large \frac{{121}}{4}\)

We have now created the perfect square trinomial \({r^2} + 11r + \Large \frac{{121}}{4}\). And, according to the last bullet of our procedure, we know that the trinomial factors as

\({r^2} + 11r + \Large \frac{{121}}{4} = {\left( {r + \Large \frac{{11}}{2}} \right)^2}\)

Congratulations, you have just completed the square.  Let’s do one more example.

Example:  Find the value of c that completes the square for \({x^2} + 38x + c\).

Solution:  We have

  • The value of \(\Large \frac{b}{2}\) is \(\Large \frac{{38}}{2} = 19\)
  • Squaring this value, we obtain 361
  • Then \(c = 361\).

We have created the perfect square trinomial \({x^2} + 38x + 361\). It factors as

\({x^2} + 38x + 361 = {\left( {x + 19} \right)^2}\)

Below you can download some free math worksheets and practice.


Downloads:
4116 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-easy

Watch below how to solve this example:

 

Downloads:
3147 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-medium

Watch below how to solve this example:

 

Downloads:
2991 x

Find the value of c that completes the square.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Completing-the-square-by-finding-the-constant-hard

Watch below how to solve this example:

 
 
 

Facebook PageYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems