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.
$9 x^{2} + 11 x + 4 = 0$

Verified step by step guidance

Identify the coefficients of the quadratic equation in the form $ax^2 + bx + c = 0$. Here, $a = 9$, $b = 11$, and $c = 4$.

Recall the formula for the discriminant: $\Delta = b^2 - 4ac$.

Substitute the values of $a$, $b$, and $c$ into the discriminant formula: $\Delta = (11)^2 - 4 \times 9 \times 4$.

Calculate the value of the discriminant (do not simplify fully as per instructions).

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Recommended video:

05:56

Introduction to Rational Equations

Related Practice

Textbook Question

Use the method described in Exercises 83–86, if applicable, and properties of absolute value to solve each equation or inequality. (Hint: Exercises 99 and 100 can be solved by inspection.) | 3x² - 14x | = 5

views

Textbook Question

Simplify each power of i. i²⁵

Textbook Question

Simplify each power of i. i²⁹

views

Textbook Question

Solve each inequality. Give the solution set using interval notation.

$- 8 > 3 x - 5 > - 12$