Explore IDNLearn.com to discover insightful answers from experts and enthusiasts alike. Ask anything and receive thorough, reliable answers from our community of experienced professionals.
Sagot :
If the line and curve intersect, it happens when
[tex]mx + 2 = x^2-5x+18[/tex]
or
[tex]x^2-(m+5)x + 16 = 0[/tex]
Recall the discriminant (denoted by ∆) of a quadratic expression:
[tex]\Delta (ax^2+bx+c) = b^2 - 4ac[/tex]
If the discriminant is positive, then the quadratic has two real roots. If it's zero, it has only one real root. If it's negative, it has two complex roots. We're interested in the third case, because that would make it so the above equation has no real roots corresponding to points of intersection in the x,y-plane.
The discriminant here is
[tex](-(m+5))^2 - 4\cdot16 = (m+5)^2-64[/tex]
Find all m such that this quantity is negative:
[tex](m+5)^2-64 < 0 \\\\ \implies (m+5)^2 < 64 \\\\ \implies \sqrt{(m+5)^2} < \sqrt{64} \\\\ \implies |m+5| < 8 \\\\ \implies -8 < m + 5 < 8 \\\\ \implies \boxed{-13 < m < 3}[/tex]
We are happy to have you as part of our community. Keep asking, answering, and sharing your insights. Together, we can create a valuable knowledge resource. IDNLearn.com has the solutions you’re looking for. Thanks for visiting, and see you next time for more reliable information.