Math Dopers: help solve this equation

Factual Questions
1

So as part of a night spent with friends at a local drinking establishment, my girlfriend proposed the following equation (oh, we do have fun):




x squared +3                  x + 17
---------------       =       --------
       7                          6

Now, she came up with this equation having the answer already. The answer is :

x=5

The question is, what method do you use to solve it? I remember little to nothing of post-Geometry math, so be gentle, please.

2

Your suggested solution does not solve the equation as written.

I’d cross multiply and solve the resulting quadratic.

3

5 most definitely does not work. In fact, you have two solutions:

x = 4.73
or
x = -3.56

Both rounded to the nearest hundredth.

To solve, you just cross multiply as the above poster said getting:

6x^2+18 = 7x+119

Combine like and get everything on one side:

6x^2-7x-101 = 0

Then use the quadratic equation to solve.

ETA: Oops! that was supposed to be 4.73 not 3.73

4

Are you sure the second numerator isn’t supposed to read “x + 19”?

5

Are you sure you copied the equation correctly? The answer would work if the 17 were replaced by 19. Otherwise you have a mess that you need to use the quadratic equation on.

6

In general, multiply both sides by the product of the denominators, subtract the right side from the left side (setting it all equal to zero), and plug all the coefficients into the quadratic equation.

There are two major errors I’m seeing: second order equations always have two roots, and neither of your equation’s roots is … in the box.

So: the positive root of the equation WOULD be in the box, if you change that “17” to a “19”

ETA: Damn! There was only one reply when I previewed!

7

Ugh . . . I mistranscribed it. Should be:



x squared + 3                x + 19
-----------------   =     ----------
      7                          6

I mistakenly put 17 instead of 19. Bleah.

8

When you suspect that the answer to a problem like this might be a small integer, the easy method is often to start trying them to see if you can find the correct one. Otherwise, you use the quadratic formula (as noted).

9

Yes, that makes more sense.

This resolves to the quadratic equation 6x[sup]2[/sup] - 7x - 115 = 0

Solve by using either the general quadratic formula or by factorising the equation to (6x + 23)(x - 5) = 0

10

Because there’s an x[sup]2[/sup] term that doesn’t disappear when you simplify, there are two solutions to the equation. 5 is the larger, and -23/6 is the smaller.

11

Thanks all for your contributions. Special credit goes to to indistinguishable, who recognized my error from the start.