For what values of k will the expression kx2+(k-2)x-(k+1) have no x term when simplified? Explain your reasoning.


We are given the quadratic expression:

\[
kx^2 + (k - 2)x - (k + 1)
\]

We want to determine the value of \( k \) such that the expression has **no \( x \)-term** when simplified.

### Step-by-step reasoning:

1. **Identify the \( x \)-term**: 
   In the expression, the term containing \( x \) is \( (k - 2)x \). For there to be no \( x \)-term, the coefficient of \( x \) must be equal to zero.

2. **Set the coefficient of \( x \) to zero**:
   The coefficient of \( x \) is \( (k - 2) \). To eliminate the \( x \)-term, we set this coefficient to zero:
   \[
   k - 2 = 0
   \]

3. **Solve for \( k \)**:
   \[
   k = 2
   \]

### Conclusion:
For \( k = 2 \), the expression will have no \( x \)-term when simplified.

To verify:
Substitute \( k = 2 \) into the original expression:
\[
2x^2 + (2 - 2)x - (2 + 1) = 2x^2 + 0x - 3 = 2x^2 - 3
\]
As expected, there is no \( x \)-term in the simplified expression.

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Do you want more details or have any questions?

Here are five related questions to explore:

1. What happens to the expression when \( k = 0 \)?
2. For what value of \( k \) will the expression be a perfect square?
3. How does the expression change for negative values of \( k \)?
4. What is the general form of a quadratic expression without an \( x \)-term?
5. How can you factor the expression for specific values of \( k \)?

**Tip**: Always identify terms by their coefficients to simplify expressions more easily.

For what values of k will the expression kx^2 + (k-2)x - (k+1) have no x term when simplified? Explain your reasoning.

Understand how to determine the value of k in the quadratic expression kx^2 + (k-2)x - (k+1) to eliminate the x-term when simplified. This problem highlights important algebraic concepts such as coefficients and solving quadratic equations, suitable for students in Grades 8-10.

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