To find the range of values of $p$ for which the curve $y = (p-1)x^2 - x + 5$ intersects the straight line $y = 2x + 3$ at two points, we need to set the two equations equal to each other and solve for $x$.

The two equations are:

1. $y = (p-1)x^2 - x + 5$
2. $y = 2x + 3$

Step 1: Set the equations equal

$(p-1)x^2 - x + 5 = 2x + 3$

Step 2: Rearrange the equation

Move all terms to one side: $(p-1)x^2 - x + 5 - 2x - 3 = 0$ Simplify: $(p-1)x^2 - 3x + 2 = 0$

Step 3: Determine the discriminant for real intersections

For the quadratic equation $(p-1)x^2 - 3x + 2 = 0$ to have two real solutions (indicating two points of intersection), the discriminant must be positive. The discriminant $\Delta$ of a quadratic equation $ax^2 + bx + c = 0$ is given by: $\Delta = b^2 - 4ac$ For our equation $(p-1)x^2 - 3x + 2 = 0$, we have:

• $a = p-1$
• $b = -3$
• $c = 2$

The discriminant is: $\Delta = (-3)^2 - 4(p-1)(2)$ $\Delta = 9 - 8(p-1)$ $\Delta = 9 - 8p + 8$ $\Delta = 17 - 8p$

Step 4: Solve for the condition for two intersections

For two real intersections, we need the discriminant to be positive: $17 - 8p > 0$

Solving for $p$: $17 > 8p$ $p < \frac{17}{8}$ Thus, the value of $p$ must satisfy: $p < \frac{17}{8} = 2.125$

Step 5: Conclusion

The curve $y = (p-1)x^2 - x + 5$ intersects the line $y = 2x + 3$ at two points if: $p < 2.125$

Final Answer:

The range of values of $p$ for which the curve intersects the line at two points is $p < 2.125$.

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Would you like further clarification or a detailed explanation of any part? Here are 5 follow-up questions:

1. How would the solution change if we required only one point of intersection?
2. Can you derive the points of intersection for a specific value of $p$?
3. What happens to the curve when $p$ is greater than 2.125?
4. How does the discriminant help us determine the nature of the solutions to a quadratic equation?
5. How would the equation change if the line had a different slope or y-intercept?

Tip: Always remember that the discriminant of a quadratic equation gives you the number and type of solutions. A positive discriminant means two real solutions, zero means one real solution, and a negative discriminant means no real solutions.