Problem 89 - Precalculus Student Solutions Manual 5th

Question

Evaluate the discriminant for each equation. Then use it to predict the number of distinct solutions, and whether they are rational, irrational, or non real complex. Do not solve the equation. $$9 x^{2}+11 x+4=0$$

Step-by-Step Solution

Answer

The discriminant is -23. The equation has two non-real complex solutions.

Step 1 - Identify coefficients

For the quadratic equation in the form of \[ax^{2} + bx + c = 0\], identify the coefficients \(a\), \(b\), and \(c\). In this case, \(a = 9\), \(b = 11\), and \(c = 4\).

Step 2 - Write the discriminant formula

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

Step 3 - Substitute the coefficients into the discriminant formula

Substitute \(a = 9\), \(b = 11\), and \(c = 4\) into the discriminant formula: \[D = 11^{2} - 4(9)(4)\].

Step 4 - Calculate the discriminant

Calculate the value of the discriminant: \(D = 121 - 144 = -23\). So, \(D = -23\).

Step 5 - Interpret the discriminant

If \(D < 0\), the equation has two non-real complex solutions. Since \(D = -23\) (which is less than 0), this means that the solutions are non-real and complex.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. Its general form is \[ ax^{2} + bx + c = 0 \] where 'a', 'b', and 'c' are coefficients with 'a' ≠ 0. This type of equation represents a parabola when graphed on the coordinate plane. Each quadratic equation can have up to two solutions, also known as roots.

The solutions can be found using various methods, such as factoring, completing the square, or using the quadratic formula. However, whether these solutions are real or complex can be predicted by calculating a specific value known as the discriminant.

Discriminant

The discriminant is a key concept in determining the nature of the roots of a quadratic equation. It is derived from the coefficients of the quadratic equation using the formula:\[ D = b^{2} - 4ac \]
Here's how the discriminant helps us understand the solutions:

• If \( D > 0\), the equation has two distinct real solutions.
• If \( D = 0\), the equation has exactly one real solution (a repeated root).
• If \( D < 0\), the equation has two complex (non-real) solutions.

To put it simply, the discriminant tells us whether the quadratic equation will intersect the x-axis (giving real solutions) or not (giving complex solutions). For example, in the equation \( 9x^{2} + 11x + 4 = 0 \), we identified \(a = 9\), \(b = 11\), and \(c = 4\). Using these values, we calculated the discriminant as:\[ D = 11^{2} - 4(9)(4) = 121 - 144 = -23 \] Since \( D = -23\) (which is less than 0), we concluded that the equation has two non-real complex solutions.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is less than 0. These solutions are not real numbers but are instead complex numbers, which have the form:\[ z = a + bi \] where 'a' and 'b' are real numbers, and 'i' is the imaginary unit, representing \( \sqrt{-1} \).
In the context of quadratic equations, when the discriminant is negative, it means the parabola does not intersect the x-axis at any point, indicating no real roots. Instead, the roots are complex and occur in conjugate pairs to maintain the real nature of the coefficients. For instance, our quadratic equation \( 9x^{2} + 11x + 4 = 0 \) had a discriminant of \( D = -23\), showing that the roots are complex.

Complex solutions to quadratic equations enrich our understanding of functions and broaden the scope of what solutions can be. Understanding how to identify these solutions is crucial for many advanced topics in mathematics, science, and engineering.