9.3 - Factoring quadratics by completing the square

The completing the square method can be used to factor any quadratic trinomial whatsoever. We assume that you have already read these sections:

Here is the procedure for factoring any quadratic trinomial using the completing the square method:
  • Manipulate the quadratic trinomial a x 2 + b x + c into completed square form using the completing the square method (click here to see the details):
  • Write the second term inside the big brackets as the square of its own square root:
  • The result is a difference of squares inside the big brackets, which can be factored using the difference of squares formula:
    A 2B 2 = (A + B) (A − B)
    In this formula let:
    The final result is that the quadratic trinomial a x 2 + b x + c can be factored as:


Notes on the above formula:


Example: Factor the expression x 2 + 6 x − 5.

Follow these steps:


Example: Factor the expression 2 x 2 + 9 x − 20.

Just for variety we will factor this one using floating point rather than exact arithmetic. Follow these steps:

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The complex case

We saw above that any quadratic trinomial a x 2 + b x + c can be factored as:
The quantity b 2 − 4 a c inside the square root is called the discriminant, and is denoted with the letter D. If D is negative then the above formula contains square roots of negative numbers. Over the real numbers this is not allowed and means that the quadratic trinomial can't be factored. But if we are doing algebra over the complex numbers then the square root of a negative number is simply an imaginary number. Thus the only complication is that the factors contain imaginary numbers.



Example: Factor the expression x 2 + 4 x + 14.

We will factor using exact (rather than floating point) arithmetic. Follow these steps: The expression is now in completed square form. At this point we can proceed in two different directions. Either way the result will be the same. The right side of this formula gives the expression in factored form. The other direction in which to proceed is this:


Example: Factor the expression 2 x 2 + 9 x + 14.

We will factor using floating point arithmetic. Follow these steps:

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