Solving an operation involving roots of a quadratic equation

Question

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 ?

Answer 1

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?"

Answer 2

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 $4$.

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Your mistake: You took $\sqrt{(x_1+x_2)^2}=x_1+x_2$

But, in this case, $x_1+x_2\neq |x_1+x_2|$

Answer 3

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 $-\frac{7}{3} + \frac{5}{3} = -\frac{2}{3}$.