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

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) \$

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 \$

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 \$

After simplification

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

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

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

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

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

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

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

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

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

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

\$ x = 7 and x = 4 \$

Option

C

Can you summarize what you’ve understood in the above steps?