6х^2-3=6^2х-7

To solve the equation \( 6x^2 - 3 = 6^2x - 7 \), let's start by simplifying the equation step by step. First, calculate \( 6^2 \): \[ 6^2 = 36 \] Now, substitute that back into the equation: \[ 6x^2 - 3 = 36x - 7 \] Next, let's rearrange the equation by moving all terms to one side: \[ 6x^2 - 36x - 3 + 7 = 0 \] This simplifies to: \[ 6x^2 - 36x + 4 = 0 \] Now, we can simplify the equation by dividing everything by 2: \[ 3x^2 - 18x + 2 = 0 \] Next, we can apply the quadratic formula to solve for \( x \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 3 \), \( b = -18 \), and \( c = 2 \). Plugging in these values: \[ x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} \] \[ x = \frac{18 \pm \sqrt{324 - 24}}{6} \] \[ x = \frac{18 \pm \sqrt{300}}{6} \] Next, simplify \( \sqrt{300} \): \[ \sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3} \] Substituting this back into the equation: \[ x = \frac{18 \pm 10\sqrt{3}}{6} \] Now, simplify: \[ x = 3 \pm \frac{5\sqrt{3}}{3} \] The solutions are: \[ x = 3 + \frac{5\sqrt{3}}{3} \quad \text{and} \quad x = 3 - \frac{5\sqrt{3}}{3} \] Thus, the final solutions are: \[ x = 3 + \frac{5\sqrt{3}}{3} \quad \text{and} \quad x = 3 - \frac{5\sqrt{3}}{3} \]