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Math Help - Expansion and Factorisation of Algebraic Expressions

  1. #1
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    Expansion and Factorisation of Algebraic Expressions

    1. If a + b = x and ab = y, express (a − b)^2 in terms of x and y.

    (A) x^2 − y (B) x2 ^− 2y (C) x^2 − 4y (D) x^2 + 2y (E) x^2 − y^2

    if i apply the formula for (a − b)^2 then i dont get any of the answers.the problem comes as to where a+b=x should be used.

    2. Simplify (−a − b) 2 − (a + b)(a − b) .

    i get ANSWER
    -(a^2+2ab+b^2)-a^2-b^2=-2a^2-2ab=-2a(a-b)

    But the actual answer is 2b^2 + 2ab.How ?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by haftakhan View Post
    1. If a + b = x and ab = y, express (a − b)^2 in terms of x and y.

    (A) x^2 − y (B) x2 ^− 2y (C) x^2 − 4y (D) x^2 + 2y (E) x^2 − y^2

    if i apply the formula for (a − b)^2 then i dont get any of the answers.the problem comes as to where a+b=x should be used.
    \displaystyle a + b = x \implies (a + b)^2 = x^2 \implies a^2 + 2ab + b^2 = x^2 \implies a^2 + 2y + b^2 = x^2

    So from the first eqaution, we get \displaystyle a^2 + b^2 = x^2 - 2y


    Hence, \displaystyle (a - b)^2 = a^2 - 2ab + b^2 = x^2 - 2y - 2y = x^2 - 4y


    2. Simplify (−a − b) 2 − (a + b)(a − b) .

    i get ANSWER
    -(a^2+2ab+b^2)-a^2-b^2=-2a^2-2ab=-2a(a-b)

    But the actual answer is 2b^2 + 2ab.How ?
    \displaystyle (-a - b)^2 - (a + b)(a - b) = (a + b)^2 - (a^2 - b^2)

    \displaystyle = a^2 + 2ab + b^2 - a^2 + b^2

    \displaystyle = 2ab + 2b^2


    the minus sign in front of a^2 + 2ab + b^2was your problem. it is not the same as what you had before. squaring gets rid of the negative
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    But where has the minus at the start has gone ?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by haftakhan View Post
    But where has the minus at the start has gone ?
    \displaystyle (-a - b)^2 = [-(a + b)]^2 = (-1)^2 (a + b)^2 = (a + b)^2
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  5. #5
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    hmm.Now i get it.
    Thanks for all the help
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