Theory of Intelligence
A-LEVEL•Mathematics•Quadratics•Question 7

Solution

🟡 Exam-style · [5 marks] Find the set of values of the constant $m$ for which the line $y = mx - 1$ does not meet the curve $y = x^2 + 2x + 3$.

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Solution

x2+2x+3=mx−1x2+(2−m)x+4=0\begin{aligned} x^2 + 2x + 3 &= mx - 1 \\ x^2 + (2 - m)x + 4 &= 0 \end{aligned}

No intersection means no real roots:

(2−m)2−16<0(2−m−4)(2−m+4)<0(−m−2)(6−m)<0(m+2)(m−6)<0\begin{aligned} (2 - m)^2 - 16 &< 0 \\ (2 - m - 4)(2 - m + 4) &< 0 \\ (-m - 2)(6 - m) &< 0 \\ (m + 2)(m - 6) &< 0 \end{aligned}

The critical values are m=−2m = -2 and m=6m = 6, and the region is between them:

−2<m<6-2 < m < 6

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