The set of all real numbers x such that $x^{2} + 2 x, 2 x + 3$ and $x^{2} + 3 x + 8$ are the sides of a triangle is............

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Solution

In any triangle sum of any two sides $\geq$ third side $∴ x \geq 5, x \geq - 11 / 3, 2 x^{2} + 3 x + 5 \geq 0$ ....(I)
sign of a quadratic expression
Note ; For all real values of x, the sign of quadratic expression is same as that of its first term provided its $Δ = - i v e$ Here $Δ = 9 - 40 = - 31 = - i v e$ Hence its sign is same as that of its first term 2 i.e. + ive
Alt. $2 x^{2} + \frac{3}{2} x + \frac{5}{2} = + i v e$
or $2 {(x + \frac{3}{4})}^{2} + \frac{5}{2} - \frac{9}{16} = + i v e$ for all real values of x.
Hence $x \geq 5$

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