Polynomial Equations

What is a perfect square?

There are many equations that can be factored into a perfect square. Any expression written in the form x2 + 2ax + a2 is a perfect square—an expression written as [something]2. To determine if an expression is a perfect square, first see if the constant term is a square number—in other words, can you take the square root of it and get an integer for an answer. If so, determine if the square root of the constant, multiplied by 2 gives the coefficient of the linear term (or the x term). If it does, the original expression may be factored into a perfect square. (Note: The above procedure only works when the coefficient of x2 is 1.)

For example, in the equation x2 + 8x + 16, the constant term (16) is already a perfect square (the square root of 16 is 4). Since 2(4) = 8, the original expression can be written as a perfect square. Because we know x2 + 2ax + a2 is a perfect square, and equals (x + a)2, by substituting the common factor 4 into the equation, we find that x2 + 8x + 16 = (x + 4)2.


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