The problem is to solve the quadratic equation $2x^2 - 11x + 14 = 0$ for $x$.

1. Problem Description

The problem is to solve the quadratic equation $2x^2 - 11x + 14 = 0$ for $x$.

2. Solution Steps

We can solve the quadratic equation $2x^2 - 11x + 14 = 0$ by factoring. We are looking for two numbers that multiply to $2 \cdot 14 = 28$ and add up to $-11$. These numbers are $-4$ and $-7$. We can rewrite the middle term $-11x$ as $-4x - 7x$. Thus,

$2x^2 - 11x + 14 = 2x^2 - 4x - 7x + 14 = 0$.

Now, we factor by grouping:

$2x(x - 2) - 7(x - 2) = 0$

$(2x - 7)(x - 2) = 0$

To find the solutions, we set each factor equal to zero:

$2x - 7 = 0$ or $x - 2 = 0$

Solving for $x$ in each case gives:

$2x = 7 \implies x = \frac{7}{2}$

$x = 2$

Therefore, the solutions are $x = \frac{7}{2}$ and $x = 2$.

3. Final Answer

$x = \frac{7}{2}, 2$