Mathematics

Mathematics

Problem

The quadratic equation (n+1)x2+2(n−2)x+7=0(n+1)x^{2}+2(n-2)x+7=0 has no real roots for certain real values of nn. Find the set of all such nn.
Intuitive explanation

A quadratic has no real roots when its discriminant is negative—that's your key condition. Here, you'll compute Δ=b2−4ac\Delta = b^2 - 4ac using the coefficients from the given equation, then solve the inequality Δ<0\Delta < 0 to find which values of nn make that happen. Don't forget to check that the equation is actually quadratic (the leading coefficient can't be zero).

Intuitive explanation

A quadratic has no real roots when its discriminant is negative—that's your key condition. Here, you'll compute Δ=b2−4ac\Delta = b^2 - 4ac using the coefficients from the given equation, then solve the inequality Δ<0\Delta < 0 to find which values of nn make that happen. Don't forget to check that the equation is actually quadratic (the leading coefficient can't be zero).

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