1

Problem

Solve the equation: \$(x + 1)/(x − 3) - (3x − 2)/(x + 2) = 2\$.

2

Step

The given equation

\$(x + 1)/(x − 3) − (3x − 2)/(x + 2) = 2\$

3

Step

To solve this equation, let’s find the least common denominator (LCD) of the rational expressions, which is (x − 3) (x + 2):

\$((x + 1)(x + 2) − (3x − 2)(x − 3))/((x − 3)(x + 2))\$ = 2

4

Step

Expanding and simplifying the numerator:

\$((x^2 + 3x + 2) − (3x^2 − 11x + 6))/((x − 3)(x + 2)) = 2\$

\$(−2x^2 + 14x − 4)/((x − 3)(x + 2)) = 2/1\$

5

Step

By using cross multiplication, and then simplified

\$- 2x^2 + 14x − 4 = 2(x − 3)(x + 2)\$

\$- 2x^2 + 14x − 4 = 2(x2 − x − 6)\$

\$- 2x^2 + 14x − 4 = 2x^2 - 2x - 12\$

6

Step

Bringing all terms to one side,then simplified

\$2x^2 - 2x - 12 + 2x^2 - 14x + 4 = 0\$

\$x^2 - 4x - 2 = 0\$

7

Formula

Now, we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula is x = \$((−b) ± \sqrt(b^2 − 4ac))/(2a)\$.

8

Step

In this case, a = 1, b = - 4, and c = - 2. Plugging these values into the quadratic formula, we have:

x = \$(−(−4) ± \sqrt((−4)^2 − 4⋅(1) (−2)))/(2⋅1)\$

x = \$(4 ± 2\sqrt(6))/2\$

x = \$2 ± \sqrt(6)\$

9

Solution

Therefore, the solutions to the original equation are x = \$2 + \sqrt(6)\$ and x = \$2 − \sqrt(6)\$.

10

Sumup

Please summarize Problem, Clue, Hint, Formula, Steps and Solution

Choices

11

Choice-A

This option is incorrect because solutions don’t match the ones we found, which were x = \$2 + \sqrt(6), 2 − \sqrt(6)\$

Wrong \$2 - \sqrt(6)\$, \$-2 - \sqrt(6)\$

12

Choice-B

This option is incorrect solution, x = \$2 + \sqrt(6)\$, but pairs it with x = \$2 − \sqrt(6)\$ . which doesn’t match the solution derived from our calculation

Wrong \$2 + \sqrt(6)\$, \$-2 - \sqrt(6)\$

13

Choice-C

This option is incorrect because it only partially matches the correct set of solutions

Wrong \$-2 + \sqrt(6)\$, \$2 - \sqrt(6)\$

14

Choice-D

This option is correct it suggests both solutions involve 2 as their base value, but with different signs for the square root term

Correct \$2 + \sqrt(6)\$, \$2 - \sqrt(6)\$

15

Answer

Option

D

16

Sumup

Please summarize choices