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What is the value of a+b?

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What is the value of a+b?

by fskilnik@GMATH » Tue Feb 26, 2019 9:48 am

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GMATH practice exercise (Quant Class 14)

What is the value of a+b ?

(1) a^2+ab−2a=91
(2) b^2+ab−2b=−28

Answer: [spoiler]____(E)__[/spoiler]
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Source: — Data Sufficiency |
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two quadratics

by GMATGuruNY » Tue Feb 26, 2019 1:32 pm

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 14)

What is the value of a+b ?

(1) a^2+ab−2a=91
(2) b^2+ab−2b=−28
Statements combined:

Adding the two equations, we get:
a² + b²+ 2ab - 2a - 2b = 63
(a+b)² - 2(a+b) = 63
(a+b)(a+b-2) = 63

Let x = a+b.
Substituting x=a+b into (a+b)(a+b-2) = 63, we get:
(x)(x-2) = 63
x² - 2x - 63 = 0
(x-9)(x+7) = 0
x=9 or x=-7

Since x=a+b, it is possible that a+b=9 or that a+b=-7.
Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.
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by fskilnik@GMATH » Wed Feb 27, 2019 6:18 am

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fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 14)

What is the value of a+b ?

(1) a^2+ab−2a=91
(2) b^2+ab−2b=−28
$$? = a + b$$
$$\left( 1 \right)\,\,a\left( {a + b - 2} \right) = 91\,\,\,\,:\,:\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {91,\, - 88} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {1,92} \right)\,\,\,\, \Rightarrow \,\,\,? = 93\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,b\left( {a + b - 2} \right) = - 28\,\,\,:\,:\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {31,\, - 28} \right)\,\,\,\, \Rightarrow \,\,\,? = 3\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 27,1} \right)\,\,\,\, \Rightarrow \,\,\,? = - 26\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\,\left( 1 \right)\left( + \right)\left( 2 \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left( {a + b - 2} \right)\left( {a + b} \right) = 63\,\,\,\, \Rightarrow \,\,\,\,{\left( {a + b} \right)^2} - 2\left( {a + b} \right) - 63 = 0$$
$$\left. \matrix{
{\rm{Sum}}:2 \hfill \cr
{\rm{Product}}: - 63\,\, \hfill \cr} \right\}\,\,\,\, \Rightarrow \,\,\,?\,\,\,\,:\,\,\,\,\,\left( {\rm{i}} \right)a + b = \,9\,\,\,\,\,{\rm{or}}\,\,\,\,\,\,\left( {{\rm{ii}}} \right)a + b\, = - 7\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left( {\rm{E}} \right)\,$$
$$\left( * \right)\,\,{\rm{viability}}\,\,\,{\rm{:}}\,\,\,\,\left\{ \matrix{
\,\left( {\rm{i}} \right)\,\,a + b = \,9\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,7a = 91\,\,\,\, \Rightarrow \,\,\,\left( {a,b} \right) = \left( {13, - 4} \right)\,\,\,\,{\rm{viable}}!\,\,\,\,\,\,\left[ {\left( 1 \right),\left( 2 \right)\,\,{\rm{ok!}}} \right] \hfill \cr
\,\left( {{\rm{ii}}} \right)\,\,a + b = \, - 7\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\, - 9a = 91\,\,\,\, \Rightarrow \,\,\,\left( {a,b} \right) = \left( { - {{91} \over 9},{{28} \over 9}} \right)\,\,\,\,{\rm{viable}}!\,\,\,\,\,\,\left[ {\left( 1 \right),\left( 2 \right)\,\,{\rm{ok!}}} \right] \hfill \cr} \right.$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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