x²−2x−35=0 .

To solve the quadratic equation \( x^2 - 2x - 35 = 0 \), we can either factor the equation or use the quadratic formula. Let's start with factoring.

We need to find two numbers that multiply to \(-35\) (the constant term) and add up to \(-2\) (the coefficient of \(x\)). The numbers \(-7\) and \(5\) satisfy these conditions since:

\[ -7 \times 5 = -35 \quad \text{and} \quad -7 + 5 = -2. \]

Thus, we can factor the quadratic equation as:

\[ (x - 7)(x + 5) = 0. \]

Setting each factor to zero gives us the solutions:

1. \( x - 7 = 0 \) \(\Rightarrow\) \( x = 7 \)
2. \( x + 5 = 0 \) \(\Rightarrow\) \( x = -5 \)

Therefore, the solutions to the equation \( x^2 - 2x - 35 = 0 \) are:

\[ x = 7 \quad \text{and} \quad x = -5. \]