Question

Select all the solutions to the equation 3x^2+9x-9=-3x^2-2x+1

Answer 1

To solve the equation \( 3x^2 + 9x - 9 = -3x^2 - 2x + 1 \), we first need to combine like terms. We can start by moving all terms to one side of the equation.

Here's the rearrangement step:

\[
3x^2 + 9x - 9 + 3x^2 + 2x - 1 = 0
\]

Now, combine the terms:

\[
(3x^2 + 3x^2) + (9x + 2x) + (-9 - 1) = 0
\]

This simplifies to:

\[
6x^2 + 11x - 10 = 0
\]

Next, we can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this case, \( a = 6 \), \( b = 11 \), and \( c = -10 \).

First, we calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = 11^2 - 4 \cdot 6 \cdot (-10) = 121 + 240 = 361
\]

Now we can apply the quadratic formula:

\[
x = \frac{-11 \pm \sqrt{361}}{2 \cdot 6} = \frac{-11 \pm 19}{12}
\]

This gives us two potential solutions:

1. For \( -11 + 19 \):

\[
x = \frac{8}{12} = \frac{2}{3}
\]

2. For \( -11 - 19 \):

\[
x = \frac{-30}{12} = -\frac{5}{2}
\]

Thus, the solutions to the equation \( 3x^2 + 9x - 9 = -3x^2 - 2x + 1 \) are:

\[
\boxed{\left\{ \frac{2}{3}, -\frac{5}{2} \right\}}
\]