Questions: Solve for (x). [3 x^2-2 x=5]

Solution Steps

To solve the quadratic equation \(3x^2 - 2x = 5\), we first rearrange it into the standard form \(ax^2 + bx + c = 0\). Then, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).

We start with the equation: \[ 3x^2 - 2x = 5 \] Rearranging it into standard form gives: \[ 3x^2 - 2x - 5 = 0 \]

The discriminant \(D\) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting \(a = 3\), \(b = -2\), and \(c = -5\): \[ D = (-2)^2 - 4 \cdot 3 \cdot (-5) = 4 + 60 = 64 \]

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] we find the two solutions: \[ x_1 = \frac{-(-2) + \sqrt{64}}{2 \cdot 3} = \frac{2 + 8}{6} = \frac{10}{6} = \frac{5}{3} \] \[ x_2 = \frac{-(-2) - \sqrt{64}}{2 \cdot 3} = \frac{2 - 8}{6} = \frac{-6}{6} = -1 \]

Final Answer