Determine the values of 
𝑚
m such that the line 
𝑦
=
𝑚
𝑥
−
4
y=mx−4 intersects the curve 
𝑦
=
−
𝑥
2
+
6
𝑥
−
8
y=−x 
2
 +6x−8 at two distinct points.

To find the values of \( m \) such that the line \( y = mx - 4 \) intersects the curve \( y = -x^2 + 6x - 8 \) at **two distinct points**, we need to solve the system of equations and analyze the discriminant of the resulting quadratic equation.

### Step 1: Set the two equations equal
We equate the two equations because they represent the points of intersection:

\[
mx - 4 = -x^2 + 6x - 8
\]

### Step 2: Rearrange the equation into a quadratic form
Move all terms to one side to form a quadratic equation:

\[
x^2 + (m - 6)x + (-4 + 8) = 0
\]

Simplifying:

\[
x^2 + (m - 6)x + 4 = 0
\]

This is the quadratic equation: 

\[
x^2 + (m - 6)x + 4 = 0
\]

### Step 3: Apply the discriminant condition
For the line to intersect the curve at two distinct points, the discriminant of the quadratic equation must be **positive**. The discriminant (\( \Delta \)) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

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

Here, \( a = 1 \), \( b = m - 6 \), and \( c = 4 \). Substituting these values into the discriminant formula:

\[
\Delta = (m - 6)^2 - 4(1)(4)
\]
\[
\Delta = (m - 6)^2 - 16
\]

For two distinct points, the discriminant must be positive:

\[
(m - 6)^2 - 16 > 0
\]

### Step 4: Solve the inequality
Solve \( (m - 6)^2 - 16 > 0 \):

\[
(m - 6)^2 > 16
\]

Take the square root of both sides:

\[
|m - 6| > 4
\]

This gives two conditions:

\[
m - 6 > 4 \quad \text{or} \quad m - 6 < -4
\]

Solving each:

1. \( m > 10 \)
2. \( m < 2 \)

### Step 5: Conclusion
The values of \( m \) such that the line intersects the curve at two distinct points are:

\[
m < 2 \quad \text{or} \quad m > 10
\]

### Would you like further details or clarifications on any part of this?

Here are some related questions:
1. What happens if \( m \) is between 2 and 10?
2. How do you determine if a line is tangent to a curve?
3. How would you solve if the discriminant equals zero?
4. Can the quadratic formula be used to find exact points of intersection?
5. What is the geometric interpretation of \( m \)?

**Tip**: Always check the discriminant for any quadratic equation when analyzing intersections, as it tells you the number of solutions (intersections).

Determine the values of m such that the line y = mx − 4 intersects the curve y = −x^2 + 6x − 8 at two distinct points.

Learn how to determine the values of m such that the line y = mx - 4 intersects the curve y = -x^2 + 6x - 8 at two distinct points. This problem involves solving systems of equations and using the discriminant to analyze quadratic equations. Suitable for Grades 11-12 students.

Math Problem Statement

Determine the values of 𝑚 m such that the line 𝑦

𝑚 𝑥 − 4 y=mx−4 intersects the curve 𝑦

Solution

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