A curve has equation $$\(y=3 x^{2}-4 x+4\)$$ and a straight line has equation $$\(y=m x+m-1\)$$, where $$\(m\)$$ is a constant. Find the set of values of $$\(m\)$$ for which the curve and the line have two distinct points of intersection. ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_w20_qp_13 Year：2020 Question No：4
Answer:
\[
3 x^{2}-4 x+4=m x+m-1 \rightarrow 3 x^{2}-(4+m) x+(5-m)(=0)
\]
\[
b^{2}-4 a c=(4+m)^{2}-4 \times 3 \times(5-m)
\]
\[
m^{2}+20 m-44
\]
\[
(m+22)(m-2)
\]
\(m>2, \quad m<-22\)
3 x^{2}-4 x+4=m x+m-1 \rightarrow 3 x^{2}-(4+m) x+(5-m)(=0)
\]
\[
b^{2}-4 a c=(4+m)^{2}-4 \times 3 \times(5-m)
\]
\[
m^{2}+20 m-44
\]
\[
(m+22)(m-2)
\]
\(m>2, \quad m<-22\)
Knowledge points:
1.1.2 find the discriminant of a quadratic polynomial and use the discriminant
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.3.5 understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations. (e.g. to determine the set of values of for which the line intersects, touches or does not meet a quadratic curve.)
Solution:
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