Question 83586
Is:
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{{{x^2-24x+48}}}
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a perfect square?  The answer is, "No, it is not."
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But how do you easily get that answer.  The method to use is called "Completing the Square."
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How do you use that method?  First, make sure that the coefficient of the {{{x^2}}} term
is 1.  [In this case it is, otherwise you would have to factor out from all the terms in
the trinomial the coefficient (multiplier) of the {{{x^2}}} term.]
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Next, forget about the +48 for the time being.  Take half of the multiplier of the x term.  That
multiplier is -24 and half of it is -12.  Then square the -12 to get + 144.
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Now we can say that {{{x^2 - 24x + 144}}} is the perfect square trinomial. [Its equivalent
form is {{{(x-12)^2}}}.] So for the given trinomial to be a perfect square, the +48 would
have to be changed to +144.
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Another way to do this (maybe even easier) is to use the discriminant of the quadratic
formula.  The quadratic formula applies to all equations of the general form:
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{{{ax^2 + bx + c = 0}}}
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The discriminant of this equation is given by {{{b^2 - 4*a*c}}}. If the discriminant
is equal to zero, the trinomial is a perfect square (the equation has two real and 
equal roots).
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By comparing your problem with the general form of the quadratic equation you can see
that a = 1, b = -24, and c = +48.  Substituting these values into the discriminant,
you get:
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{{{(-24)^2- 4*1*48}}}
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This simplifies to:
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{{{576 - 4*48 = 576 -192 = 384}}}
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Since the discriminant does not equal zero, the given trinomial is not a perfect square.
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However, suppose that the given trinomial had been {{{x^2 - 24x + 144}}}. In this case
a = 1, b = -24, and c = 144.
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Substitute these values into the discriminant and you get:
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{{{b^2 - 4*a*c = (-24)^2 - 4*1*144 = 576 - 576 = 0}}}
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And since the discriminant now equals zero, this trinomial is a perfect square. [As we
noted earlier the trinomial {{{x^2 -24x+144}}} is equivalent to {{{(x-12)^2}}}.]
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Hope these couple of ways help you to understand your problem better.