The equation of a curve is y equals x squared - 3x + 4. The equation of a line is y + 2x = k, where k is a constant. Find the value of k for which is the intersection that has exactly one root

To find the value of \( k \) for which the curve \( y = x^2 - 3x + 4 \) and the line \( y + 2x = k \) intersect at exactly one point, we need to set the equations equal to each other and find conditions where there is exactly one solution.

### Step 1: Write the equations

1. The equation of the curve is:

   \[
   y = x^2 - 3x + 4
   \]

2. The equation of the line is:

   \[
   y + 2x = k \Rightarrow y = k - 2x
   \]

### Step 2: Set the equations equal to each other

Substitute \( y \) from the line equation into the curve equation:

\[
x^2 - 3x + 4 = k - 2x
\]

### Step 3: Rearrange into a standard quadratic equation

Rearrange this equation to set it equal to zero:

\[
x^2 - 3x + 4 = k - 2x
\]

\[
x^2 - 3x + 2x + 4 - k = 0
\]

\[
x^2 - x + (4 - k) = 0
\]

### Step 4: Use the discriminant to find when there is exactly one solution

For a quadratic equation \( ax^2 + bx + c = 0 \) to have exactly one solution, its discriminant (\(\Delta\)) must be zero. The discriminant is given by:

\[
\Delta = b^2 - 4ac
\]

For our quadratic equation \( x^2 - x + (4 - k) = 0 \), the coefficients are:

- \( a = 1 \)
- \( b = -1 \)
- \( c = 4 - k \)

Substituting these into the discriminant formula:

\[
\Delta = (-1)^2 - 4(1)(4 - k)
\]

\[
\Delta = 1 - 4(4 - k)
\]

\[
\Delta = 1 - 16 + 4k
\]

\[
\Delta = 4k - 15
\]

### Step 5: Set the discriminant to zero for exactly one root

To find the value of \( k \) that results in exactly one intersection (i.e., one root), set the discriminant to zero:

\[
4k - 15 = 0
\]

\[
4k = 15
\]

\[
k = \frac{15}{4}
\]

\[
k = 3.75
\]

### Conclusion

The value of \( k \) for which the curve \( y = x^2 - 3x + 4 \) and the line \( y + 2x = k \) intersect at exactly one point is:

\[
k = 3.75
\]

Would you like to see more details or have any other questions?

### Related Questions:

1. What are the coordinates of the intersection point when \( k = 3.75 \)?
2. How does the discriminant change for different values of \( k \)?
3. What are the solutions for the quadratic equation when \( k = 3.75 \)?
4. How would the problem change if the curve's equation was different?
5. What does it mean graphically when the curve and line have exactly one intersection?

**Tip:** Using the discriminant is a powerful method to determine the nature of the roots of a quadratic equation, including when two curves intersect at exactly one point.

Learn how to find the value of k for which the curve y = x^2 - 3x + 4 and the line y + 2x = k intersect at exactly one point. Detailed step-by-step solution with discriminant analysis. Suitable for students in Grades 10-12.

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