Реши

To solve the quadratic equation \(5x^2 - 11x - 9 = 3x^2 - 11x + 9\), follow these steps:

1. Simplify the Equation:

   First, let's move all terms to one side to get a standard quadratic form. Subtract \(3x^2\) and \(-11x\) from both sides:

   \[ 5x^2 - 11x - 9 - 3x^2 + 11x - 9 = 0 \]

   Simplify the terms:

   \[ (5x^2 - 3x^2) + (-11x + 11x) - 9 - 9 = 0 \]

   \[ 2x^2 - 18 = 0 \]

2. Solve for \(x\):

   Next, solve for \(x\). First, isolate \(x^2\):

   \[ 2x^2 = 18 \]

   Divide both sides by 2:

   \[ x^2 = 9 \]

3. Find the Square Roots:

   Solve for \(x\) by taking the square root of both sides:

   \[ x = \pm \sqrt{9} \]

   \[ x = \pm 3 \]

   So, the solutions are \(x = 3\) and \(x = -3\).

4. Check the Solutions:

   Substitute \(x = 3\) and \(x = -3\) back into the original equation to verify.

   • If \(x = 3\): \[ 5(3)^2 - 11(3) - 9 = 3(3)^2 - 11(3) + 9 \] \[ 45 - 33 - 9 = 27 - 33 + 9 \] \[ 3 = 3 \quad (\text{True}) \]

   • If \(x = -3\): \[ 5(-3)^2 - 11(-3) - 9 = 3(-3)^2 - 11(-3) + 9 \] \[ 45 + 33 - 9 = 27 + 33 + 9 \] \[ 69 = 69 \quad (\text{True}) \]

The solutions \(x = 3\) and \(x = -3\) satisfy the equation.