Line Curve Intersection F91D02

1. **State the problem:** Find the values of $m$ such that the line $y=mx - 5$ does not intersect the curve $y=x^2 - 5x + 4$. 2. **Set the equations equal to find intersection points:** $$mx - 5 = x^2 - 5x + 4$$ 3. **Rearrange to form a quadratic equation in $x$:** $$x^2 - 5x + 4 - mx + 5 = 0$$ $$x^2 - (5 + m)x + 9 = 0$$ 4. **Recall the condition for no intersection:** The quadratic equation has no real solutions if its discriminant $\Delta < 0$. 5. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-(5 + m))^2 - 4 \times 1 \times 9 = (5 + m)^2 - 36$$ 6. **Set the inequality for no real roots:** $$(5 + m)^2 - 36 < 0$$ $$(5 + m)^2 < 36$$ 7. **Solve the inequality:** $$-6 < 5 + m < 6$$ $$-6 - 5 < m < 6 - 5$$ $$-11 < m < 1$$ **Final answer:** The line $y=mx - 5$ does not intersect the curve $y=x^2 - 5x + 4$ for values of $m$ in the interval $$(-11, 1)$$.