A question from my book:
$3x^2 + 7x + 5 = 0$ So, $\sqrt{(x_1^2 + 2x_1x_2 + x_2^2)} + x_1x_2 = ?$
Options: A) $4$ B) $5$ C) $6$ D) $7$ E) $8$
It's looking too easy, my answer is $-\frac{2}3$, but it does not exist in options and my book says that right answer is $4$.
Please help, where is my mistake ?
Hint:
$$\sqrt{x_1^2+2x_1x_2+x_2^2}+x_1x_2=\sqrt{(x_1+x_2)^2}+x_1x_2=|x_1+x_2|+x_1x_2$$
Now, use Vieta's formulas. The answer is $4$.
Your mistake: You took $\sqrt{(x_1+x_2)^2}=x_1+x_2$
But, in this case, $x_1+x_2\neq |x_1+x_2|$
You have $$\sqrt{x_1^2+2x_1x_2+x_2^2} + x_1x_2 = |x_1+x_2|+x_1x_2 = \left| -\frac{7}{3} \right| + \frac{5}{3} = 4$$
I think your mistake comes from ignoring that the square root of something is always nonnegative, since $-\frac{7}{3} + \frac{5}{3} = -\frac{2}{3}$.
It is difficult to tell you what mistake you made if you don't show your workings. However you might want to note that by convention the square root is a positive value and the sum of the roots of your original equation is negative.
Now you have edited to show $-\frac 23$ rather than $-\frac 12$ that looks likely to be the source of the difference. To avoid such problems in future ask "why have they put a square root in there, what difference does it make?"
$x_1 + x_2 = -\frac{7}{3}$ $x_1x_2 = \frac{5}{3}$ $\frac{5}{3} - \frac{7}{3} = -\frac{2}{3}$
– Oğuz İsmayil uysal Apr 07 '15 at 09:41