1

Problem

Solve the equation: \$(2/x) + 3/(x^2)\$ = 5.

2

Step

The given equation

\$(2/x) + 3/(x^2)\$ = 5.

3

Step

To solve this equation, let’s first make a substitution. Let’s set a variable y, equal to \$x^(−1)\$. So,

y = \$1/x\$

4

Step

Now we can rewrite the equation in terms of y, rearranging the equation, we get:

\$2y + 3y^2 = 5\$

\$3y^2 + 2y − 5 = 0\$

5

Formula

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

6

Step

In this case, a = 3, b = 2, and c = - 5. Plugging these values into the quadratic formula, we have:

y = \$(-(-2) ± \sqrt(2^2 − 4(3)( − 5)))/(2(3))\$

y = \$(−2 ± 8)/6\$

7

Step

So, we have two possible solutions for y then we get x value

\$y_1 = (8 − 2)/6\$ = \$6/6\$ = 1 and

\$y_2 = (- 8 - 2 )/6\$ = \$-(10)/6\$ = \$- 5/3\$

\$1/x\$ = 1 and \$1/x\$ = \$- 5/3\$

\$x_1 = 1/1\$ and \$x_2 = 1/(- 5/3)\$

\$x_1 = 1\$ and \$x_2 = - 3/5\$

8

Solution

Therefore, the solutions to the original equation are \$x_1 = 1 "and" x_2 = −3/5\$

9

Sumup

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

Choices

10

Choice-A

This option is incorrect, because the calculation showed that after rearranging and applying the quadratic formula, these values did not satisfy the equation

Wrong 1, \$-5/3\$

11

Choice-B

This option is correct because after applying the quadratic formula or another method, these were the values of x that satisfied the original equation

Correct 1, \$-3/5\$

12

Choice-C

This option is incorrect because after solving the equation, we find that x = −1 does not satisfy the given equation.

Wrong 1, -1

13

Choice-D

This option is incorrect, because in this case, since option B correctly identifies the solutions, option D would not be the right choice

Wrong None of these above

14

Answer

Option

B

15

Sumup

Please summarize choices