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Problem 696

AMC 10/12, early questions

Algebra Difficulty 3.9 Multiple choice UNIONE MATEMATICA ITALIANA SCUOLA NORMALE SUPERIORE DI PISA GARA di SECONDO LIVELLO · Italy

Of the three roots $a, b, c$ of the polynomial $2x^{3}-7x^{2}-2x+12$ we know that $4a=3(b+c)$ . What is the value of $a+bc$ ?

Pick one

Official solution

Solution:

The answer is (A). We have $a+b+c=\frac{7}{2}$ and $abc=-6$ . Hence $4a+4b+4c=14$ and, using the data of the problem, $3(b+c)+4b+4c=14$ , from which $b+c=2$ . Therefore $a=\frac{3}{2}$ and $bc=-4$ , so $a+bc=\frac{3}{2}-4=-\frac{5}{2}$ .

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