For the equation \(2x^2+3x+\lambda=0\) to have real and distinct roots, which condition on \(\lambda\) is correct?

Correct answer: \(\lambda<\frac{9}{8}\)

A quadratic equation \(ax^2+bx+c=0\) has real and distinct roots when its discriminant satisfies \(D=b^2-4ac>0\). Here, \(a=2\), \(b=3\), and \(c=\lambda\), so \(D=3^2-4(2)(\lambda)=9-8\lambda\). Thus, \(9-8\lambda>0\), which gives \(\lambda<\frac{9}{8}\). When \(\lambda=\frac{9}{8}\), the roots are equal, so option B is not correct. Exam tip: For two distinct real roots, always use the condition \(D>0\).