Question

find undefined terms, denominators ldc, rewrite the equation with ldc, simplify show final equations that has no denoinator, solve for x, plug in the values for x n the original equations to check for extraneous solutions, state finals solutions

Answer 1

I don’t see a specific equation from you — below I give the exact steps you asked for and a complete worked example so you can follow the same process for any rational equation.

Example equation
(x/(x − 2)) + (2/(x + 3)) = 5/(x^2 + x − 6)

1) Find undefined values (where any denominator = 0)
- Denominators: x − 2, x + 3, x^2 + x − 6
- x − 2 = 0 → x = 2
- x + 3 = 0 → x = −3
So x = 2 and x = −3 are undefined and must be excluded from any solution set.

2) Factor denominators and find the least common denominator (LCD)
- x^2 + x − 6 factors as (x − 2)(x + 3)
- LCD = (x − 2)(x + 3)

3) Multiply both sides by the LCD to clear denominators
Multiply the whole equation by (x − 2)(x + 3). Each term simplifies:
- x/(x−2) × LCD = x(x + 3)
- 2/(x+3) × LCD = 2(x − 2)
- 5/(x^2 + x − 6) × LCD = 5

So we get:
x(x + 3) + 2(x − 2) = 5

4) Simplify to get an equation with no denominators
Expand and combine like terms:
x^2 + 3x + 2x − 4 = 5
x^2 + 5x − 4 = 5
x^2 + 5x − 9 = 0

5) Solve for x
Use the quadratic formula:
x = [−5 ± sqrt(5^2 − 4·1·(−9))]/2
= [−5 ± sqrt(25 + 36)]/2
= [−5 ± sqrt(61)]/2

So the candidate solutions are x = (−5 + sqrt(61))/2 and x = (−5 − sqrt(61))/2.

6) Check for extraneous solutions by plugging into the original equation
First confirm neither candidate equals 2 or −3 (they do not), so they don’t make any denominator zero.

You can also check numerically (or symbolically). Numerically:
- x1 = (−5 + sqrt(61))/2 ≈ 1.405124838 → LHS ≈ RHS ≈ −1.908766
- x2 = (−5 − sqrt(61))/2 ≈ −6.405124838 → LHS ≈ RHS ≈ 0.174766

Both satisfy the original equation and are not excluded by denominators, so neither is extraneous.

Final answer (exact)
x = (−5 ± sqrt(61))/2

(Exclude x = 2 and x = −3 as they make denominators zero.)

If you have a different equation you want solved, paste it and I’ll run the same steps on that one.