(2х-3)²=11х-19

To solve the equation \((2x - 3)^2 = 11x - 19\), we start by expanding the left side of the equation. 1. Expand \((2x - 3)^2\): \[ (2x - 3)(2x - 3) = 4x^2 - 12x + 9 \] 2. Now, rewrite the equation: \[ 4x^2 - 12x + 9 = 11x - 19 \] 3. Move all terms to one side of the equation: \[ 4x^2 - 12x + 9 - 11x + 19 = 0 \] \[ 4x^2 - 23x + 28 = 0 \] 4. Now we can solve this quadratic equation using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 4\), \(b = -23\), and \(c = 28\). First, calculate the discriminant (\(b^2 - 4ac\)): \[ (-23)^2 - 4 \cdot 4 \cdot 28 = 529 - 448 = 81 \] 5. Since the discriminant is positive, we have two real solutions. Now plug the values into the formula: \[ x = \frac{23 \pm \sqrt{81}}{2 \cdot 4} \] \[ x = \frac{23 \pm 9}{8} \] 6. This gives us two solutions: - For \(x = \frac{23 + 9}{8}\): \[ x = \frac{32}{8} = 4 \] - For \(x = \frac{23 - 9}{8}\): \[ x = \frac{14}{8} = \frac{7}{4} \] Thus, the solutions to the equation \((2x - 3)^2 = 11x - 19\) are: \[ x = 4 \quad \text{and} \quad x = \frac{7}{4} \]