Pearson Algebra 2 Common Core, 2011
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Pearson Algebra 2 Common Core, 2011 View details
6. Completing the Square
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Exercise 9 Page 237

We can solve an equation of the form where and are real numbers, by completing the square. Notice that we can always rewrite any quadratic equation in this form by isolating the variable terms and dividing by the coefficient of if needed. Let's see an example.
Now we need to ensure that we have an expression that is a perfect square trinomial. Then, we will be able to factor it as the square of a binomial and take the square root on both sides to find the solutions. Let's recall the expansion of the square of a binomial.
By adding the appropriate constant, we can obtain a perfect square trinomial. Let's compare the right-hand side of this formula to the left-hand side of our expression.
Notice that the parameter of our expression corresponds to the factor in the expansion for the square of the binomial. We can find the quantity corresponding to that we would have to add from this relationship by solving for first and then squaring both sides.
This means that by adding the term we can get a perfect square trinomial! Let's try this with the example equation we had at the beginning, Notice that here
Then, adding to both sides of will let us rewrite the left-hand side as the square of a binomial.
As we can see, the solutions for this equation are and

Summary of the Procedure

As we saw above, we can complete the square by following the next steps.

  1. We isolate the terms containing the variable
  2. We divide both sides by the coefficient of
  3. We identify the coefficient of the linear term as
  4. We add to both sides.