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

Algebra ->  Equations -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!

+7x%5E2+%2B+4x+=+3
+7x%5E2+%2B+4x+-+3=0
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 7%2Ax%5E2%2B4%2Ax-3=0 ( notice a=7, b=4, and c=-3)





x+=+%28-4+%2B-+sqrt%28+%284%29%5E2-4%2A7%2A-3+%29%29%2F%282%2A7%29 Plug in a=7, b=4, and c=-3




x+=+%28-4+%2B-+sqrt%28+16-4%2A7%2A-3+%29%29%2F%282%2A7%29 Square 4 to get 16




x+=+%28-4+%2B-+sqrt%28+16%2B84+%29%29%2F%282%2A7%29 Multiply -4%2A-3%2A7 to get 84




x+=+%28-4+%2B-+sqrt%28+100+%29%29%2F%282%2A7%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-4+%2B-+10%29%2F%282%2A7%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-4+%2B-+10%29%2F14 Multiply 2 and 7 to get 14


So now the expression breaks down into two parts


x+=+%28-4+%2B+10%29%2F14 or x+=+%28-4+-+10%29%2F14


Lets look at the first part:


x=%28-4+%2B+10%29%2F14


x=6%2F14 Add the terms in the numerator

x=3%2F7 Divide


So one answer is

x=3%2F7




Now lets look at the second part:


x=%28-4+-+10%29%2F14


x=-14%2F14 Subtract the terms in the numerator

x=-1 Divide


So another answer is

x=-1


So our solutions are:

x=3%2F7 or x=-1




so, solutions are x=3%2F7 or x=-1
check if
I. The sum of the solutions is positive.
3%2F7%2B%28-1%29=3%2F7-1=0.43-1=-0.57

II. The product of the solutions is negative.
%283%2F7%29%28-1%29=-3%2F7

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

answer:
b) II only

Answer by richard1234(7193) About Me  (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.