SOLUTION: Solve using quadratic formula. 1 over X^2 minus 3 equals 8 over x The solution is -4 over 3 (plus or minus) the square root of 19 over 3. I don't understand how.

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Question 1027738: Solve using quadratic formula.
1 over X^2 minus 3 equals 8 over x
The solution is -4 over 3 (plus or minus) the square root of 19 over 3. I don't understand how.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
me neither.

i graphed the factors and got the following graph.

the factors are (3x + 4 - sqrt(19)) * (3x + 4 + sqrt(19))

when you multiply those factors out, you get 9x^2 + 24x - 3.

i took that and then did the following:

start with 9x^2 + 24x - 3

subtract 24x from both sides to get:

9x^2 - 3 = -24x

factor the left side of the equation to get:

3 * (3x^2 - 1) = -24x

divide both sides of the equation to get:

3x^2 - 1 = -8x

divide both sides of the equation by 8x to get:

(3x^2 - 1) / 8x = -1

divide both sides of the equation by (3x^2 - 1) to get:

1 / 8x = -1 / (3x^2 - 1)

i then added -1 / (3x^2 - 1) to both sides of this equation to get:

1 / 8x + 1 / (3x^2 - 1) = 0

the roots of this equation are the same roots as the equation of 9x^2 + 24x - 3 = 0.

i graphed 3 equations to show you that the roots are the same for all 3.

the first 2 equations i graphed are:

y = (3x + 4 - sqrt(19)) * (3x + 4 + sqrt(19))

y = 9x^2 + 24x - 3.

these 2 equations are equivalent and show up as the same line on the graph.

the roots are x = -2.786 and x = .12

that graph is shown below:

the other equation i graphed is:

y = 1 / 8x + 1 / (3x^2 - 1)

that graph is shown below:

while this graph looks very different, the roots are the same.

the closest i could get to what you were showing with the roots you showed is:

start with:

9x^2 + 24x - 3 = 0

subtract 24x from both sides of the equation to get:

9x^2 - 3 = -24x

factor out the 3 on the left hand side to get:

3 * (3x^2 - 1) = -24x

divide both sides of the equation by 3 to get:

3x^2 - 1 = -8x

divide both sides of the equation by x and divide both sides of the equation by (3x^2 - 1) to get:

1/x = -8 / (3x^2 - 1)

that's the closest i could get to what you were showing.

it's not it.

what you are showing does not result in the roots you are showing.

i took the equation of 1/x = -8 / (3x^2 - 1) and added the right side of the equation to to both sides of the equation to get:

1/x + 8/(3x^2-1) = 0

the roots of that equation are the same, even though that equation looks different also.

the graph of y = 1/x + 8/(3x^2-1) is shown below.

Answer by MathTherapy(10551) (Show Source):
You can put this solution on YOUR website!

Solve using quadratic formula.
1 over X^2 minus 3 equals 8 over x
The solution is -4 over 3 (plus or minus) the square root of 19 over 3. I don't understand how.

-------- Multiplying by LCD, x²

a = 3 b = 8 c = - 1
Quadratic equation formula:
------- Substituting 3 for a, 8 for b, and - 1 for c