Maths-
General
Easy

Question

Find the simplified Quotient , and state the domain.
fraction numerator begin display style fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 4 x minus 21 end fraction end style over denominator begin display style fraction numerator 2 x squared plus 5 x minus 3 over denominator 3 x plus 9 end fraction end style end fraction

hintHint:

The expansions of certain identities are:
open parentheses x squared minus a squared close parentheses equals left parenthesis x plus a right parenthesis left parenthesis x minus a right parenthesis
left parenthesis x plus a right parenthesis left parenthesis x plus b right parenthesis equals x squared plus left parenthesis a plus b right parenthesis x plus a b
We are asked to find the quotient and state the domain of the expression.

The correct answer is: the domain is; (-∞, 7) ∪ (7, ∞)


    Step 1 of 3:
    The given expression is fraction numerator begin display style fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 4 x minus 21 end fraction end style over denominator begin display style fraction numerator 2 x squared plus 5 x minus 3 over denominator 3 x plus 9 end fraction end style end fraction.
    Take the reciprocal of the second expression and them multiply it with the first expression. This works same as division;

    fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 4 x minus 21 end fraction cross times fraction numerator 3 x plus 9 over denominator 2 x squared plus 5 x minus 3 end fraction
    Step 2 of 3:
    Simplify the expression and cancel out the common factors;
    fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 4 x minus 21 end fraction cross times fraction numerator 3 x plus 9 over denominator 2 x squared plus 5 x minus 3 end fraction equals fraction numerator 2 x squared plus 5 x minus 3 over denominator x squared minus 7 x plus 3 x minus 21 end fraction cross times fraction numerator 3 left parenthesis x plus 3 right parenthesis over denominator 2 x squared plus 5 x minus 3 end fraction
    equals fraction numerator 1 over denominator x left parenthesis x minus 7 right parenthesis plus 3 left parenthesis x minus 7 right parenthesis end fraction cross times fraction numerator 3 left parenthesis x plus 3 right parenthesis over denominator 1 end fraction
    equals fraction numerator 1 over denominator left parenthesis x plus 3 right parenthesis left parenthesis x minus 7 right parenthesis end fraction cross times fraction numerator 3 left parenthesis x plus 3 right parenthesis over denominator 1 end fraction
    equals fraction numerator 3 over denominator x minus 7 end fraction
    Thus, the quotient is : fraction numerator 3 over denominator x minus 7 end fraction.
    Step 3 of 3:
    The domain of the expression should exclude the value for which the denominator attains a zero value.
    x - 7 = 0
                                                                                                          x = 7
    Thus, the domain is; (-∞, 7) ∪ (7, ∞)

    Division of an expression by zero is not defined. That is why; we exclude the values of zero for the denominator.

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