To solve the equation 6 / x^2 - 9 - 1 / x - 3 = 1, we need to combine the fractions first:
6 / x^2 - 9 - 1 / x - 3 = 1
6 / (x^2 - 9) - 1 / (x - 3) = 1
(6(x - 3) - (x^2 - 9)) / (x^2 - 9)(x - 3) = 1
(6x - 18 - x^2 + 9) / (x^2 - 9)(x - 3) = 1
(-x^2 + 6x - 9) / (x^3 - 9x - 3x^2 + 27) = 1
(-x^2 + 6x - 9) / (x^3 - 3x^2 - 9x + 27) = 1
-x^2 + 6x - 9 = x^3 - 3x^2 - 9x + 27
x^3 - 3x^2 - 9x + 27 + x^2 - 6x + 9 = 0
x^3 - 2x^2 - 15x + 36 = 0
Now, we can factorize the equation as follows:
(x - 3)(x + 3)(x - 4) = 0
Therefore, the solutions to the equation are x = 3, x = -3, and x = 4.
Checking these values back in the original equation, we get 6 / 3^2 - 9 - 1 / 3 - 3 = 1, which simplifies to 0 = 1 (false),
- for x = -3, we get 6 / (-3)^2 - 9 - 1 / (-3) - 3 = 6 / 9 - 9 + 1 / -3 - 3 = 0 - 9 + (-1) - 3 = -13, which is not equal to 1.
- for x = 4, we get 6 / 4^2 - 9 - 1 / 4 - 3 = 6 / 16 - 9 - 1 / 4 - 3 = 6 / 16 - 9 - 1 / 1 = 6 / 16 - 9 - 1 = 9 - 9 - 1 = -1, which is not equal to 1.
Therefore, none of the solutions x = 3, x = -3, and x = 4 satisfy the equation. The correct answer according to the options provided is none of the mentioned options.
solve the equation. check the solution
6 / x^2 - 9 - 1 / x - 3 = 1
A. -4
B. 2
C. -1 +- square root 73 / 2
D. 3 or - 4
3 answers
Solving Rational Equations
To solve the equation, we need to first get a common denominator. The equation is:
6 / x^2 - 9 - 1 / x - 3 = 1
Combining the fractions, we get:
(6(x - 3) - (x^2 - 9)) / (x^2 - 9)(x - 3) = 1
(6x - 18 - x^2 + 9) / (x^2 - 9)(x - 3) = 1
(-x^2 + 6x - 9) / (x^2 - 9)(x - 3) = 1
-x^2 + 6x - 9 = (x^2 - 9)(x - 3)
-x^2 + 6x - 9 = x^3 - 3x^2 - 9x + 27
Rearranging terms, we get:
x^3 - 2x^2 - 15x + 36 = 0
Now, we need to find the values of x that satisfy this equation. One way to do this is by factoring or using numerical methods.
By factoring, we find that the equation can be rewritten as:
(x - 3)(x + 3)(x - 4) = 0
This implies that x = 3, x = -3, or x = 4.
Checking these values in the original equation:
For x = 3,
6 / 3^2 - 9 - 1 / 3 - 3 = 2 - 9 - 1 / 0 = 0. So, x = 3 is not a solution.
For x = -3,
6 / (-3)^2 - 9 - 1 / (-3) - 3 = 6 / 9 - 9 - 1 / -6 = 6 / 9 - 9 + 1 / 6 = 2/3 - 9 - 1/6 = 2/3 - 54/6 - 1/6 = -52/6 = -26/3. So, x = -3 is not a solution.
For x = 4,
6 / 4^2 - 9 - 1 / 4 - 3 = 6 / 16 - 9 - 1 / 1 = 6 / 16 - 9 - 1 = 3/8 - 9 - 1 = 3/8 - 72/8 - 8/8 = -77/8. So, x = 4 is not a solution.
Therefore, none of the options provided of -4, 2, -1 + sqrt(73) / 2, and 3 or -4 are correct solutions to the equation.
6 / x^2 - 9 - 1 / x - 3 = 1
Combining the fractions, we get:
(6(x - 3) - (x^2 - 9)) / (x^2 - 9)(x - 3) = 1
(6x - 18 - x^2 + 9) / (x^2 - 9)(x - 3) = 1
(-x^2 + 6x - 9) / (x^2 - 9)(x - 3) = 1
-x^2 + 6x - 9 = (x^2 - 9)(x - 3)
-x^2 + 6x - 9 = x^3 - 3x^2 - 9x + 27
Rearranging terms, we get:
x^3 - 2x^2 - 15x + 36 = 0
Now, we need to find the values of x that satisfy this equation. One way to do this is by factoring or using numerical methods.
By factoring, we find that the equation can be rewritten as:
(x - 3)(x + 3)(x - 4) = 0
This implies that x = 3, x = -3, or x = 4.
Checking these values in the original equation:
For x = 3,
6 / 3^2 - 9 - 1 / 3 - 3 = 2 - 9 - 1 / 0 = 0. So, x = 3 is not a solution.
For x = -3,
6 / (-3)^2 - 9 - 1 / (-3) - 3 = 6 / 9 - 9 - 1 / -6 = 6 / 9 - 9 + 1 / 6 = 2/3 - 9 - 1/6 = 2/3 - 54/6 - 1/6 = -52/6 = -26/3. So, x = -3 is not a solution.
For x = 4,
6 / 4^2 - 9 - 1 / 4 - 3 = 6 / 16 - 9 - 1 / 1 = 6 / 16 - 9 - 1 = 3/8 - 9 - 1 = 3/8 - 72/8 - 8/8 = -77/8. So, x = 4 is not a solution.
Therefore, none of the options provided of -4, 2, -1 + sqrt(73) / 2, and 3 or -4 are correct solutions to the equation.
1 answer