Step: 1

To solve the nonlinear equation \$3x^2 + 12x - 15 = 0\$, you can use the quadratic formula:

\$ x = (-b ± \sqrt(b^2 - 4ac)) / (2a) \$

In the given equation, a = 3, b = 12, and c = -15. Plug these values into the quadratic formula:

⇒\$ x = (-12 ± \sqrt((12)^2 - 4 times 3 times (-15))) / (2 times 3) \$

Simplify the expression under the square root:

\$ x = (- 12 ± \sqrt(144 + 180)) / 6 \$

\$ x = (- 12 ± \sqrt(324)) / 6 \$

\$ x = (- 12 ± 18) / 6 \$

Now, you have two possible solutions:

1.For the positive square root:

\$ x_1 = (- 12 + 18) / 6 = 6/6 = 1\$

2.For the negative square root:

\$ x_2 = (- 12 - 18) / 6 = - (30)/6 = - 5\$

Therefore, the solutions to the nonlinear equation \$3x^2 + 12x - 15 = 0\$ are x = 1 and x = - 5.

Explanation: