find the set of values for k for which the line y=x+k is tangent to y=x(power of 2)+x-2

To find the set of values for \( k \) such that the line \( y = x + k \) is tangent to the parabola \( y = x^2 + x - 2 \), we need to follow these steps:

### 1. **Set the equations equal to each other** (since at the point of tangency, the \( y \)-coordinates will be equal):

\[
x + k = x^2 + x - 2
\]

### 2. **Rearrange the equation to form a quadratic equation**:

\[
0 = x^2 + x - 2 - (x + k)
\]

Simplifying:

\[
0 = x^2 + x - 2 - x - k
\]

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

\[
x^2 = k + 2
\]

Thus, the equation simplifies to:

\[
x^2 = k + 2.
This

Find the set of values for k for which the line y = x + k is tangent to y = x^2 + x - 2.

Learn how to determine the set of values for k such that the line y = x + k is tangent to the parabola y = x^2 + x - 2. This problem explores key algebraic concepts, such as solving quadratic equations and applying the tangency condition. Suitable for high school students in Grades 10-12.

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