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Given that the quadratic

(r + 1) x^{2} + 2 r x + (r - 3) = 0

has no real solutions, find the range of possible values of $r$ .

Your answer should be of the form $r < α$ for some $α \in Q$ . Give the value of $α$ as an exact decimal.

Hint

b^{2} - 4 a c < 0

Solution

Since the equation has no real solutions, we have

\begin{aligned} b^{2} - 4 a c & < 0 \\ 4 r^{2} - 4 (r + 1) (r - 3) & < 0 \\ 4 r^{2} - 4 r^{2} + 8 r + 12 & < 0 \\ 2 r + 3 & < 0 \\ r & < - \frac{3}{2} \end{aligned}