Questions: Radicals and Quadratic Equations Applying the quadratic formula: Exact answers Use the quadratic formula to solve for x. 3 x^2-9 x+4=0 (If there is more than one solution, separate them with commas.) x=

Solution Steps

To solve the quadratic equation \(3x^2 - 9x + 4 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).

Given the quadratic equation: \[ 3x^2 - 9x + 4 = 0 \] We identify the coefficients: \[ a = 3, \quad b = -9, \quad c = 4 \]

The discriminant \(\Delta\) is calculated as: \[ \Delta = b^2 - 4ac \] Substituting the values: \[ \Delta = (-9)^2 - 4 \cdot 3 \cdot 4 = 81 - 48 = 33 \]

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substituting the values: \[ x = \frac{-(-9) \pm \sqrt{33}}{2 \cdot 3} = \frac{9 \pm \sqrt{33}}{6} \]

We calculate the two solutions: \[ x_1 = \frac{9 + \sqrt{33}}{6} \approx 2.457 \] \[ x_2 = \frac{9 - \sqrt{33}}{6} \approx 0.5426 \]

Final Answer