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 tex2html_wrap_inline155 . We will actually consider the equivalent problem of finding the roots, the solutions of the equation

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Move the constant term to the other side of the equation:

displaymath140

The magic trick of this method is to exploit the binomial formula:

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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:

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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:

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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):

displaymath144

Next we take square roots of both sides, but be careful: there are two possible cases:

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In both cases tex2html_wrap_inline161 . We are done, once we solve the two equations for x.

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are the two roots of our polynomial. Consequently, our polynomial factors as follows:

eqnarray18


Another example.

Let us try to factor tex2html_wrap_inline183 . We will again consider the equivalent problem of finding the roots, the solutions of the equation

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In this example, the leading coefficient (the number in front of the tex2html_wrap_inline185 ) is 2, and thus not equal to 1; we can fix that by dividing by 2:

displaymath166

Move the constant term to the other side of the equation:

displaymath167

The magic trick of this method is to exploit the binomial formula:

displaymath141

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:

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But the third term of the binomial formula does not show up in our equation; we make it show up by adding tex2html_wrap_inline191 to both sides of our equation:

displaymath170

We have "completed the square"! Now we use the binomial formula to simplify the left side of our equation (also simplifying the right side):

displaymath171

The rest is easy: we take square roots of both sides, but be careful: there are two possible cases:

displaymath172

In both cases tex2html_wrap_inline193 . We are done, once we solve the two equations for x.

displaymath173

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!)

eqnarray73


A last example "for the road".

Let us factor tex2html_wrap_inline213 . We will again consider the equivalent problem of finding the roots, the solutions of the equation

displaymath197

Move the constant term to the other side of the equation:

displaymath198

The magic trick of this method is to exploit the binomial formula:

displaymath141

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:

displaymath200

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:

displaymath201

We have "completed the square"! Now we use the binomial formula to simplify the left side of our equation (also simplifying the right side):

displaymath202

The rest is easy: we take square roots of both sides, but be careful: there are two possible cases:

displaymath203

In both cases tex2html_wrap_inline221 . We are done, once we solve the two equations for x.

displaymath204

are the two roots of our polynomial. Here is the factorization of our polynomial.

eqnarray102


Exercise 1.

Find the roots of the polynomial tex2html_wrap_inline225 by completing the square.

Answer.

Exercise 2.

Factor the polynomial tex2html_wrap_inline227 by completing the square.

Answer.

There are more exercises on the next page dealing with the quadratic formula.

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Helmut Knaust

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