The product of the real roots of the equation ${ \left| 2x+3 \right| }^{ 2 }-3\left| 2x+3 \right| +2=0$ is
- $\displaystyle\frac{5}{4}$
- $\displaystyle\frac{5}{2}$
- $5$
- $2$
Let u = |2x+3|. u^2 - 3u + 2 = 0 => (u-1)(u-2) = 0. So |2x+3| = 1 or |2x+3| = 2. Case 1: 2x+3 = 1 => x = -1; 2x+3 = -1 => x = -2. Case 2: 2x+3 = 2 => x = -0.5; 2x+3 = -2 => x = -2.5. Roots are -1, -2, -0.5, -2.5. Product = (-1)(-2)(-0.5)*(-2.5) = 2 * 1.25 = 2.5 = 5/2.
Let the absolute value expression |2x + 3| be represented by y, which transforms the equation into y squared minus 3y plus 2 equals 0. Factoring this quadratic equation gives (y minus 1)(y minus 2) equals 0, yielding y equals 1 and y equals 2. Reverting to x gives the equations |2x + 3| equals 1 and |2x + 3| equals 2, which yield the real roots negative 1, negative 2, 1/2, and negative 5/2. The product of all the real roots is (negative 1) times (negative 2) times (1/2) times (negative 5/2), which equals 5/2. The result is 5/2.