Problem 73 - A Graphical Approach to College Algebra

Question

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$9 x^{2}+11 x+4=0$$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Identify the parts of the quadratic equation

The given quadratic equation is in the standard form \( ax^2 + bx + c = 0 \). For the equation \( 9x^2 + 11x + 4 = 0 \), we identify \( a = 9 \), \( b = 11 \), and \( c = 4 \).

Step 2: Write the discriminant formula

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \).

Step 3: Calculate the discriminant

Substitute the values of \( a \), \( b \), and \( c \) into the formula: \( \Delta = 11^2 - 4 \times 9 \times 4 \). First calculate \( 11^2 = 121 \) and \( 4 \times 9 \times 4 = 144 \). Therefore, \( \Delta = 121 - 144 = -23 \).

Step 4: Analyze the discriminant

A negative discriminant (\( \Delta < 0 \)) indicates that the quadratic equation has no real solutions. It specifically means there are two complex conjugate solutions.

Key Concepts

Understanding Real Solutions

When working with quadratic equations, one significant aspect is determining whether the equation can have real solutions. A real solution refers to a solution with no imaginary component. Here's a brief overview of how the discriminant, \(\Delta\), helps us find this:

• When \(\Delta > 0\), there are two distinct real solutions. These solutions can either be rational or irrational depending on whether the square root of \(\Delta\) is a perfect square.
• When \(\Delta = 0\), there is exactly one real solution, which is also a repeated or double solution. This value is always rational.
• When \(\Delta < 0\), there are no real solutions; instead, the solutions are complex.

Understanding this can guide you in analyzing quadratic equations quickly without actually solving them. Just compute \(\Delta\) and check its sign to know the nature and number of real solutions.

Basics of a Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two, generally written as \(ax^2 + bx + c = 0\). In this form:

• \(a\) is the coefficient of the \(x^2\) term and must not be zero, or else the equation isn't quadratic.
• \(b\) is the coefficient of the \(x\) term.
• \(c\) is the constant term or free term.

Understanding each part of the quadratic equation helps when using the discriminant to determine the nature of solutions. In our example \(9x^2 + 11x + 4 = 0\), we have identified \(a = 9\), \(b = 11\), and \(c = 4\). Knowing these values is crucial for evaluating the discriminant accurately.

Delving into Complex Solutions

Complex solutions occur when the discriminant of a quadratic equation is less than zero (\(\Delta < 0\)). In such cases, the solutions are not real but complex conjugates of each other. Here's what you need to know about complex solutions:

• They appear in pairs because complex roots occur as conjugates. If one solution is \(a + bi\), the other is \(a - bi\).
• These solutions include an imaginary unit \(i\), where \(i^2 = -1\).

For our example \(9x^2 + 11x + 4 = 0\), since \(\Delta = -23\), the solutions involve imaginary numbers. This means they are located on the complex plane, not the real number line, making them distinct from real solutions.