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The quadratic equation

Source: UMD 2023 I #14

October 23, 2023
quadraticsUMDalgebra

Problem Statement

Let m≠−1m \neq -1 be a real number. Consider the quadratic equation (m+1)x2+4mx+m−3=0. (m + 1)x^2 + 4mx + m - 3 =0. Which of the following must be true?
 \rm(I) Both roots of this equation must be real.
 \rm(II) If both roots are real, then one of the roots must be less than −1.-1.
 \rm(III) If both roots are real, then one of the roots must be larger than 1.1.
a. Only (I)b. (I) and (II)c. Only (III)d. Both (I) and (III)e. (I),(II), and (III) \mathrm a. ~ \text{Only} ~(\mathrm I)\rm \qquad \mathrm b. ~(I)~and~(II)\qquad \mathrm c. ~Only~(III) \qquad \mathrm d. ~Both~(I)~and~(III) \qquad \mathrm e. ~(I), (II),~and~(III)
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