Please help me solve for x. I'm stumped!?

Question

I need to solve for x. Here's the problem:

((4)/(x-3)) = ((3)/(x-4))+(1/2)

I keep coming up with x squared minus 9x + 26 = 0. From there, I've applied the quadratic formula and keep ending up with the square root of a negative number, which is obviously impossible. Can someone shed some light on this for me? Thanks a million!

Answer 1

(4/(x - 3)) = (3/(x - 4)) + (1/2)

Multiply everything by 2(x - 3)(x - 4)

4(2(x - 4)) = 3(2(x - 3)) + ((x - 3)(x - 4))

8(x - 4) = 6(x - 3) + (x^2 - 4x - 3x + 12)
8x - 32 = 6x - 18 + x^2 - 7x + 12
8x - 32 = x^2 - x - 6
x^2 - 9x + 26 = 0

x = (-b ± sqrt(b^2 - 4ac))/(2a)

x = (-(-9) ± sqrt((-9)^2 - 4(1)(26)))/(2(1))
x = (9 ± sqrt(81 - 104))/2
x = (9 ± sqrt(-23))/2
x = (9 ± isqrt(23))/2

x = (1/2)(9 + isqrt(23)) or (1/2)(9 - isqrt(23))

sqrt of a negative number is i or +i

getting the sqrt of a negative number isn't IMPOSSIBLE, its IMPROBABLE. The different is that IMPOSSIBLE means "something that can't or shouldn't be", IMPROBABLE means "it's unthinkable, but not impossible".

Lets take this for ex: Man traveling to the moon, was once thought to be IMPOSSIBLE, but scientist thought of traveling to the moon as IMPROBABLE, but not IMPOSSIBLE. That is how most mathmeticians and scientists think of most things. Just look at all the IMPOSSIBLE now IMPROBABLE technology we now have.

Answer 2

I'm getting x^2 - 9x +26 as well. And as you say, there is no x-intercept using the quadratic formula. If you don't know about complex numbers (specifically, i - the square root of -1), you're done here and the answer is "no solution," since there is nothing that can be plotted on the graph as an x-intercept. (Remember that these quadratics are parabolas, and not all parabolas will intersect with the x-axis.)

If you know how to use i, you can go a little further - I get x equals 9 +/- the square root of -23 over 2. If I remember college math right, that works out to 9/2 +/- sqrt 23 over 2 times i.

Answer 3

4= (3x-9)/(x-4) + (x-3)/2\
4x-16= 3x-9 + (x-3)(x-4)/2

x-7= (x^2-7x +12)/2
2x-14=x^2-7x+12
x^2-9x+26=0

inside the square root you get:
81-4(26)=81- 104 which is , as you said a negative number,
this means that the quadratic equation does not have a solution,
and therefore neither does the original equation.

Answer 4

I got the same thing you did. Have you learned about imaginary numbers yet?


I got (9+/_sqrt(-23))/2, which is the same as (9 +/- i*sqrt 23)/2

Answer 5

This would not follow a quadratic equation. In its form:
ax2 +bx+c = 0

This appears to be a simple balanced equation

Answer 6

multiply by (x-3):
4 = 3x-9/x-4 + 1/2
put 1/2 on left:
3.5 = 3x-9/x-4
multiply by 2:
7 = 6x-18/x-4
multiply by x-4
7x-28= 6x-18
x=10