Question

The curve y= (k+2)x^2 meets the line y=(k+3)x +3. Find the range of possible values of k.


Could somebody explain how they get to their answer, too?

Skills: Simultaneous Equations, Discriminant, Quadratic Inequality & Rearranging Quadratics

Answer 1

`(k+2)x^2=(k+3)x+3\\ (k+2)x^2-(k+3)x-3=0\\ \Delta\geq0 \wedge k\\ot=-2\\ \Delta=(-(k-3))^2-4\cdot(k+2)\cdot(-3)\\ \Delta=(-k+3)^2+12k+24\\ \Delta=k^2-6k+9+12k+24\\ \Delta=k^2+6k+33\\ k^2+6k+33\geq0\\ \Delta=6^2-4\cdot1\cdot33=36-132=-9<0 \Rightarrow k\in \mathbb{R}\\\\\ k\in \mathbb{R} \wedge k\\ot =-2\\ \boxed{k\in \mathbb{R}\setminus\{-2\}}`