SOLUTION: If 7x^2 + 4x = 3 , which of the following is (are) true? I. The sum of the solutions is positive. II. The product of the solutions is negative. III. Both solutions are integer

Question 463547: If 7x^2 + 4x = 3 , which of the following is (are) true?
I. The sum of the solutions is positive.
II. The product of the solutions is negative.
III. Both solutions are integers.
a) I only
b) II only
c) III only
d) I and II only
e) II and III only
Found 2 solutions by MathLover1, richard1234:
Answer by MathLover1(20849) (Show Source): You can put this solution on YOUR website!

Solved by pluggable solver: Quadratic Formula

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=7, b=4, and c=-3

Square 4 to get 16

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 7 to get 14

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator

Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator

Divide

So another answer is

So our solutions are:

or

so, solutions are or
check if
I. The sum of the solutions is positive.

II. The product of the solutions is negative.

III. Both solutions are integers.
no, only one solution is integer

answer:
b) II only

Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!

Know Vieta's formulas, which say that if you have a polynomial

Then the sum of the roots is and the product of the roots is (plus, a whole bunch of other identities can be derived but we won't need them here).

Here, the sum of the solutions is -4/7, which is not positive, so I is not true. The product of the solutions is (-3)/7, negative, so II is true. We automatically know III cannot be true since if both solutions were integers, their sum would be an integer, but we already know I is not true. Hence, only II is true; answer is B.

For certain math problems, it can help to find a slick, fast solution. Here we were able to do it without even finding the roots of the quadratic.
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