To solve the equation \( 6x^2 - 5x - 2 = -6x^2 \), we first need to bring all terms to one side of the equation.
- Add \( 6x^2 \) to both sides:
\[ 6x^2 + 6x^2 - 5x - 2 = 0 \]
This simplifies to:
\[ 12x^2 - 5x - 2 = 0 \]
- Now we can apply the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 12 \), \( b = -5 \), and \( c = -2 \).
- We first calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 - 4ac = (-5)^2 - 4(12)(-2) = 25 + 96 = 121 \]
- Now we substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-(-5) \pm \sqrt{121}}{2(12)} \]
This simplifies to:
\[ x = \frac{5 \pm 11}{24} \]
- Now we calculate the two possible values for \( x \):
For the positive case:
\[ x = \frac{5 + 11}{24} = \frac{16}{24} = \frac{2}{3} \]
For the negative case:
\[ x = \frac{5 - 11}{24} = \frac{-6}{24} = -\frac{1}{4} \]
- Therefore, the solutions to the equation \( 12x^2 - 5x - 2 = 0 \) are:
\[ x = \frac{2}{3} \quad \text{and} \quad x = -\frac{1}{4} \]
These are the solutions in simplest form.