(2х-1)2-9х2=-35

To solve the equation \((2x - 1)^2 - 9x^2 = -35\), let's first simplify the equation step by step.

1. Start with the original equation: \[ (2x - 1)^2 - 9x^2 = -35 \]

2. Expand \((2x - 1)^2\): \[ (2x - 1)(2x - 1) = 4x^2 - 4x + 1 \] So, the equation becomes: \[ 4x^2 - 4x + 1 - 9x^2 = -35 \]

3. Combine like terms: \[ (4x^2 - 9x^2) - 4x + 1 = -35 \] \[ -5x^2 - 4x + 1 = -35 \]

4. Move \(-35\) to the left side: \[ -5x^2 - 4x + 1 + 35 = 0 \] \[ -5x^2 - 4x + 36 = 0 \]

5. Multiply the whole equation by \(-1\) to make the leading coefficient positive: \[ 5x^2 + 4x - 36 = 0 \]

6. Now, apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 5\), \(b = 4\), and \(c = -36\):

   • Calculate the discriminant \(b^2 - 4ac\): \[ 4^2 - 4 \cdot 5 \cdot (-36) = 16 + 720 = 736 \]

7. Now substitute back into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{736}}{2 \cdot 5} \]

8. Simplify \(\sqrt{736}\): \[ \sqrt{736} = \sqrt{16 \cdot 46} = 4\sqrt{46} \]

9. Substitute for \(\sqrt{736}\): \[ x = \frac{-4 \pm 4\sqrt{46}}{10} = \frac{-2 \pm 2\sqrt{46}}{5} \]

10. Thus, the solutions for \(x\) are: \[ x = \frac{-2 + 2\sqrt{46}}{5} \quad \text{and} \quad x = \frac{-2 - 2\sqrt{46}}{5} \]

So the final answers are: \[ x_1 = \frac{-2 + 2\sqrt{46}}{5}, \quad x_2 = \frac{-2 - 2\sqrt{46}}{5} \]