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

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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: \(\Delta = (5)^2 - 4 \times 3 \times 2\).

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 \(\Delta > 0\), there are two distinct real roots; if \(\Delta = 0\), there is one real root (a repeated root); if \(\Delta < 0\), the roots are nonreal complex numbers. Also, if \(\Delta\) is a perfect square, the roots are rational; otherwise, they are irrational.

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주요 개념

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Discriminant of a Quadratic Equation

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac for an equation ax² + bx + c = 0. It determines the nature and number of solutions without solving the equation.

Number and Nature of Solutions Based on the Discriminant

If the discriminant is positive, there are two distinct real solutions; if zero, one real repeated solution; if negative, two nonreal complex solutions. This helps classify the roots quickly.

Rational vs. Irrational Solutions

When the discriminant is a perfect square, the solutions are rational numbers. If it is positive but not a perfect square, the solutions are irrational. This distinction helps understand the exact form of the roots.

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