Nature of Roots

A quadratic equation has no real roots when its discriminant satisfies Δ < 0. Here, Δ = [−2(a + 1)]² − 4(a² + 4a + 5) = −8(a + 2). Thus, −8(a + 2) < 0 gives a + 2 > 0, so a > −2. When a = −2, Δ = 0, giving two equal real roots. Exam tip: simplify the discriminant completely before applying its sign condition.

Here, \(a=m+4\), \(b=-2(m+2)\), and \(c=m\). The condition \(m\ne-4\) ensures that \(a\ne0\), so the equation is genuinely quadratic. Its discriminant is \(\Delta=b^2-4ac=4(m+2)^2-4m(m+4)=16\). Since \(\Delta>0\), the roots are real and distinct. Equal roots would require \(\Delta=0\), so option B is incorrect. Exam tip: \(\Delta>0\), \(\Delta=0\), and \(\Delta<0\) indicate distinct real, equal real, and non-real roots, respectively.

A quadratic equation has no real roots when its discriminant satisfies \(D<0\). Thus, \((r-2)^2-9<0\), or \((r-2)^2<9\). This gives \(-3<r-2<3\), and hence \(-1<r<5\). At the endpoints \(r=-1\) and \(r=5\), \(D=0\), so the equation has one repeated real root; therefore option C is not correct. Exam tip: use \(D<0\) for no real roots, \(D=0\) for equal real roots, and \(D>0\) for two distinct real roots.