(2x-1)²-9x²=-35

Let's solve the equation step by step.

Given:

\[ (2x - 1)^2 - 9x^2 = -35 \]

First, expand \((2x - 1)^2\):

\[ (2x - 1)^2 = 4x^2 - 4x + 1 \]

Now, substitute this back into the equation:

\[ 4x^2 - 4x + 1 - 9x^2 = -35 \]

Combine like terms:

\[ 4x^2 - 9x^2 - 4x + 1 = -35 \] \[ -5x^2 - 4x + 1 = -35 \]

Now, add 35 to both sides:

\[ -5x^2 - 4x + 1 + 35 = 0 \] \[ -5x^2 - 4x + 36 = 0 \]

To make calculations easier, we can multiply the entire equation by -1:

\[ 5x^2 + 4x - 36 = 0 \]

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

Calculate the discriminant first:

\[ b^2 - 4ac = 4^2 - 4(5)(-36) = 16 + 720 = 736 \]

Now substitute back into the quadratic formula:

\[ x = \frac{-4 \pm \sqrt{736}}{2(5)} \] \[ x = \frac{-4 \pm \sqrt{736}}{10} \]

Next, simplify \(\sqrt{736}\):

\[ \sqrt{736} = \sqrt{16 \times 46} = 4\sqrt{46} \]

So, substitute this back into the formula:

\[ x = \frac{-4 \pm 4\sqrt{46}}{10} \] \[ x = \frac{-2 \pm 2\sqrt{46}}{5} \]

Thus, the solutions for \(x\) are:

\[ x = \frac{-2 + 2\sqrt{46}}{5} \quad \text{and} \quad x = \frac{-2 - 2\sqrt{46}}{5} \]

These are the final results. If you need further assistance, feel free to ask!