# Quadratic Polynomials: Completing the Square. If guessing does not work, "completing the square" will do the job.

#### A first example.

Let us try to factor . We will actually consider the equivalent problem of finding the roots, the solutions of the equation Move the constant term to the other side of the equation: The magic trick of this method is to exploit the binomial formula: If we look at the left side of the equation we want to solve, we see that it matches the first two terms of the binomial formula if b=1. Let's write down the binomial formula for b=1: But the third term of the binomial formula does not show up in our equation; we make it show up by force by adding 1 to both sides of our equation: This trick is called "completing the square"! Now we use the binomial formula to simplify the left side of our equation (also adding 7+1=8): Next we take square roots of both sides, but be careful: there are two possible cases: In both cases . We are done, once we solve the two equations for x. are the two roots of our polynomial. Consequently, our polynomial factors as follows: #### Another example.

Let us try to factor . We will again consider the equivalent problem of finding the roots, the solutions of the equation In this example, the leading coefficient (the number in front of the ) is 2, and thus not equal to 1; we can fix that by dividing by 2: Move the constant term to the other side of the equation: The magic trick of this method is to exploit the binomial formula: If we look at the left side of the equation we want to solve, we see that it matches the first two terms of the binomial formula if b=3/2. Let's write down the binomial formula for b=3/2: But the third term of the binomial formula does not show up in our equation; we make it show up by adding to both sides of our equation: We have "completed the square"! Now we use the binomial formula to simplify the left side of our equation (also simplifying the right side): The rest is easy: we take square roots of both sides, but be careful: there are two possible cases: In both cases . We are done, once we solve the two equations for x. are the two roots of our polynomial. Here is the factorization of our polynomial. Be careful: We have to multiply our two "standard" factors by 2 (the leading term we divided by in the beginning!) #### A last example "for the road".

Let us factor . We will again consider the equivalent problem of finding the roots, the solutions of the equation Move the constant term to the other side of the equation: The magic trick of this method is to exploit the binomial formula: If we look at the left side of the equation we want to solve, we see that it matches the first two terms of the binomial formula if b=-3. You always take half of the term in front of the x. Let's write down the binomial formula for b=-3: But the third term of the binomial formula does not show up in our equation; we make it show up by adding 9 to both sides of our equation: We have "completed the square"! Now we use the binomial formula to simplify the left side of our equation (also simplifying the right side): The rest is easy: we take square roots of both sides, but be careful: there are two possible cases: In both cases . We are done, once we solve the two equations for x. are the two roots of our polynomial. Here is the factorization of our polynomial. #### Exercise 1.

Find the roots of the polynomial by completing the square.

#### Exercise 2.

Factor the polynomial by completing the square.

There are more exercises on the next page dealing with the quadratic formula. [Back] [Next]
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