A line with equation y equals MX - 6 intersects with the curve with equation y equals 2x - 4x + 3. Find the possible values of the constant m.

We are given two equations:

1. Line equation: \( y = mx - 6 \)
2. Curve equation: \( y = 2x - 4x + 3 \) (which I believe should be \( y = 2x^2 - 4x + 3 \), as this looks like a quadratic expression).

### To find the possible values of \( m \), let's follow these steps:

1. **Set the equations equal to each other**, since at the points of intersection, the \( y \)-values are the same.

   \[
   mx - 6 = 2x^2 - 4x + 3
   \]

2. **Rearrange the equation** to set it equal to 0:

   \[
   2x^2 - 4x + 3 - mx + 6 = 0
   \]

   Simplifying the terms:

   \[
   2x^2 + (-4 - m)x + 9 = 0
   \]

   Now, we have a quadratic equation in the form \( 2x^2 + (-4 - m)x + 9 = 0 \).

3. **Analyze the discriminant** of this quadratic equation to find the possible values of \( m \). The discriminant, \( \Delta \), is given by:

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

   Here, \( a = 2 \), \( b = -4 - m \), and \( c = 9 \).

   Substitute these values into the discriminant formula:

   \[
   \Delta = (-4 - m)^2 - 4(2)(9)
   \]

   Simplifying:

   \[
   \Delta = (16 + 8m + m^2) - 72
   \]

   \[
   \Delta = m^2 + 8m - 56
   \]

4. **Determine the number of solutions**. For the line to intersect the curve, the quadratic equation must have real solutions. This happens when the discriminant is non-negative:

   \[
   m^2 + 8m - 56 \geq 0
   \]

5. **Solve the inequality**:

   First, solve the equality \( m^2 + 8m - 56 = 0 \) using the quadratic formula:

   \[
   m = \frac{-8 \pm \sqrt{8^2 - 4(1)(-56)}}{2(1)}
   \]

   \[
   m = \frac{-8 \pm \sqrt{64 + 224}}{2}
   \]

   \[
   m = \frac{-8 \pm \sqrt{288}}{2}
   \]

   \[
   m = \frac{-8 \pm 12\sqrt{2}}{2}
   \]

   \[
   m = -4 \pm 6\sqrt{2}
   \]

6. **Interpret the result**: The two possible values of \( m \) where the line intersects the curve are \( m = -4 + 6\sqrt{2} \) and \( m = -4 - 6\sqrt{2} \).

Would you like further details on any of these steps or have any questions? Here are some related questions:

1. How do you determine the discriminant of a quadratic equation?
2. What is the importance of the discriminant in solving quadratic equations?
3. Can you explain how to solve quadratic inequalities?
4. What happens when the discriminant is exactly zero?
5. How do different values of \( m \) change the line's slope?

**Tip**: Always check the discriminant to determine how many real solutions a quadratic equation has!

A line with equation y equals mx - 6 intersects with the curve with equation y equals 2x - 4x + 3. Find the possible values of the constant m.

Discover how to find the possible values of m for the intersection of a line y = mx - 6 and a quadratic curve y = 2x^2 - 4x + 3. This detailed solution explains key concepts like quadratic equations, discriminant analysis, and solving inequalities. Ideal for students in Grades 9-11.

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