Question - Simplification of a Rational Expression
Solution:
To simplify the given expression, he can factor the denominator and check if there are common factors between numerator and denominator that he can cancel out:
The denominator is an expression that looks like the difference of squares, which can be factored as:
\((2x-1)^2 - 3(x^2-1) = (2x-1)^2 - 3(x+1)(x-1)\)
He should expand the squares and multiply out the terms to simplify:
\((2x-1)^2 = (2x-1)(2x-1) = 4x^2 - 4x + 1\)
\(3(x+1)(x-1) = 3(x^2 - 1) = 3x^2 - 3\)
Subtracting the second expression from the first:
\(4x^2 - 4x + 1 - (3x^2 - 3) = 4x^2 - 4x + 1 - 3x^2 + 3\)
Simplifying:
\(x^2 - 4x + 4\)
He then recognizes this as a perfect square trinomial which can be factored as:
\(x^2 - 4x + 4 = (x-2)^2\)
So the expression simplifies to:
\(\frac{x}{(x-2)^2}\)
He notes that there are no common factors to cancel out with the numerator, hence this is the simplified form of the given rational expression.
Tutor Solution:
The expression given is:
\[ y = \frac{x}{(2x-1)^2 - 3(x^2-1)} \]First, let's expand the square and the second term in the denominator:
\[ (2x-1)^2 = 4x^2 - 4x + 1 \] \[ -3(x^2-1) = -3x^2 + 3 \]Now, let's combine these terms:
\[ (2x-1)^2 - 3(x^2-1) = (4x^2 - 4x + 1) - (3x^2 - 3) \] \[ = 4x^2 - 4x + 1 - 3x^2 + 3 \] \[ = 4x^2 - 3x^2 - 4x + 1 + 3 \] \[ = x^2 - 4x + 4 \]We now have the simplified form of the denominator, so let's rewrite our original equation:
\[ y = \frac{x}{x^2 - 4x + 4} \]Notice that the denominator can be factored further because it is a perfect square trinomial:
\[ x^2 - 4x + 4 = (x - 2)^2 \]So now the expression simplifies to:
\[ y = \frac{x}{(x - 2)^2} \]This is the simplified form of the expression, and there doesn't seem to be further simplification possible given that the numerator and the denominator do not have common factors to cancel out.
Therefore, the final result is:
\[ y = \frac{x}{(x - 2)^2} \]CamTutor
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