1

Problem

Solve the nonlinear equation \$ 3x^2 - 8x + 5 = 0\$.

2

Step

The given equation is

\$3x^2 - 8x + 5 = 0\$

3

Formula

The quadratic formula provides the solutions for x in equations of the form \$ax^2 + bx + c = 0\$: \$ x = (-b ± \sqrt(b^2 - 4ac)) / (2a) \$.

4

Step

Analyze the equation \$3x^2 - 8x + 5 = 0\$, then plug its values to the given formula:

a = 3, b = - 8, and c = 5

\$ x = (-(-8) ± \sqrt((-8)^2 - 4 \times 3 \times 5)) / (2 \times 3) \$

\$ x = (8 ± \sqrt(64 - 60)) / 6 \$

5

Step

Make it simplified and here, we have two possible solution are:

\$ x = (8 ± \sqrt4) / 6 \$

\$ x = (8 ± 2) / 6 \$

\$ x = (8 + 2) / 6 \$ and \$ x = (8 - 2) / 6 \$

6

Step

After simplification:

\$ x = (\cancel(10)^5) / (\cancel(6)^3)\$ and \$ x = (\cancel6^1) / (\cancel6^1) \$

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

7

Solution

Therefore, the solutions to the non-linear equation \$ 3x^2 - 8x + 5 = 0\$ are \$ x = 5/3 and x = 1 \$.

8

Sumup

Please summarize steps

Choices

9

Choice-A

This option states that \$x = 2/3\$​ and x = −3, which are not the correct roots of the equation

Wrong \$ x = 2/3 and x = -3 \$

10

Choice-B

This option is flawed as it wrongly considers \$x = - 1/2\$ as a solution, inconsistent with actual solutions

Wrong \$ x = - 1/2 and x = 1 \$

11

Choice-C

This option is correct because it correctly states the solutions as \$x = 5/3\$​ and x = 1

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

12

Choice-D

This option is wrong; the roots, x = 7 and x = 4, don’t satisfy the equation. Hence, incorrect

Wrong \$ x = 7 aand x = 4 \$

13

Answer

Option

C

14

Sumup

Please summarize choices