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.