Question

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) 3x2 + 5x + 2 = 0

Accepted Answer

Identify the coefficients of the quadratic equation $$3x^2 + 5x + 2 = 0$. Here, a = 3$, b = 5$, and c = 2$. Recall the formula for the discriminant of a quadratic equation ax^2$ + bx + c = 0$$, which is given by $$\Delta = b^2$ - 4ac. $$Substitute the values of $$a$$, $$b$$, and $$c$$ into the discriminant formula: $$\Delta = (5)^2$ - 4 	imes 3 	imes 2. $$Calculate the value of the discriminant expression (but do not simplify to the final number as per instructions). Use the value of the discriminant to determine the nature of the roots: if $$\Delta \u003e 0$$, there are two distinct real roots; if $$\Delta = 0$$, there is one real root (a repeated root); if $$\Delta \u003c 0$$, the roots are nonreal complex numbers. Also, if $$\Delta is a $$perfect square, the roots are rational; otherwise, they are irrational.

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) 3x² + 5x + 2 = 0

Verified video answer for a similar problem:

Key Concepts

Discriminant of a Quadratic Equation

Number and Nature of Solutions Based on the Discriminant

Rational vs. Irrational Solutions