Question - Solving Rational Expressions with Factoring
mathsolving rational expressionsfactoring in equationscommon factor in expressionsdifference of squares
Solution:
To solve the equation $$ \frac{3x^2 - 3}{x - 2x - 3} $$, you first simplify the denominator:$$ x - 2x - 3 = -x - 3 $$So the equation now is:$$ \frac{3x^2 - 3}{-x - 3} $$Both the numerator and the denominator have a common factor which can be factored out. In the numerator, you can factor out a 3, and in the denominator, you can factor out a -1:$$ \frac{3(x^2 - 1)}{-(x + 3)} $$Notice that $$ x^2 - 1 $$ is a difference of squares, which can be factored as $$ (x + 1)(x - 1) $$:$$ \frac{3(x + 1)(x - 1)}{-(x + 3)} $$Depending on what you need to do next, the equation can be left in its factored form, or if you wish to simplify further, you can divide numerator and denominator by -1, which will change the signs:$$ \frac{-3(x + 1)(x - 1)}{x + 3} $$Now the equation is fully simplified. If this is an expression that you are simplifying, that would be the final result. If you were solving for x, you would look if the denominator equals to zero because that would indicate where the expression is undefined. In this case, you cannot have $$ x + 3 = 0 $$, which simplifies to $$ x \neq -3 $$.
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