Equivalent Equations

Algebra Level 2

If a + b = 1 a + b = 1 , and a 2 − b 2 = − 3 a^{2} - b^{2} = -3 ; what is the value of ( a − b ) (a-b)


The answer is -3.

Den Onelle Dujali by Den Onelle Dujali , 23, Philippines

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2 solutions

Den Onelle Dujali
Oct 16, 2018

This actually makes use of special products, so we factor a 2 − b 2 a^{2} - b^{2} = -3 to ( a + b ) ( a − b ) (a + b)(a - b) = -3. From the given information, ( a + b ) = 1 (a + b) = 1 , our equation becomes 1 ( a − b ) = − 3 1(a - b) = -3 , then we simplify it as ( a − b ) = − 3 (a - b) = -3 , which is the answer.

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Blan Morrison
Oct 16, 2018

Relevant wiki: Difference Of Squares

a 2 − b 2 = − 3 a^2-b^2=-3 ⟹ ( a + b ) ( a − b ) = − 3 \implies(a+b)(a-b)=-3 ⟹ ( a − b ) = − 3 \implies (a-b)=-3 β ⌈ ∣ ⌉ \beta_{\lceil \mid \rceil}

Note that you also could have solved for a a and b b . Here are their values: a + b = 1 a+b=1 a − b = − 3 a-b=-3 2 a = − 2 2a=-2 a = − 1 ; b = 2 a=-1;~b=2

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