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.) See Example 9.
$9x^2+11x+4=0$

Verified step by step guidance

1. Identify the coefficients of the quadratic equation in the form \(ax^2 + bx + c = 0\). Here, \(a = 9\), \(b = 11\), and \(c = 4\).
2. Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\).
3. Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (11)^2 - 4 \times 9 \times 4\).
4. Calculate the value of the discriminant (do not simplify fully as per instructions).
5. Use the value of the discriminant to determine the nature of the roots: if \(\Delta > 0\) and a perfect square, roots are rational and distinct; if \(\Delta > 0\) but not a perfect square, roots are irrational and distinct; if \(\Delta = 0\), roots are real and equal; if \(\Delta < 0\), roots are nonreal complex numbers.

Key Concepts

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 Type 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 as real or complex.

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 describe the exact nature of the roots.