Problem 35 - Algebra 2

Question

Complete parts a–c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. \(4 x^{2}+7=9 x\)

Step-by-Step Solution

Answer

Discriminant: -31; Two complex roots: \( x = \frac{9 \pm i\sqrt{31}}{8} \).

Step 1: Write the Equation in Standard Form

The first step in solving a quadratic equation is to write it in the standard form, which is \( ax^2 + bx + c = 0 \). We start with the equation \( 4x^2 + 7 = 9x \). Subtract \( 9x \) from both sides to get:\[ 4x^2 - 9x + 7 = 0 \]

Step 2: Calculate the Discriminant

The discriminant in a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( b^2 - 4ac \). In our equation, \( a = 4 \), \( b = -9 \), and \( c = 7 \). Calculate the discriminant:\[ (-9)^2 - 4 \times 4 \times 7 = 81 - 112 = -31 \]

Step 3: Describe the Number and Type of Roots

The sign of the discriminant tells us the number and type of roots:- If the discriminant is positive, there are two distinct real roots.- If it is zero, there is one real root (a repeated root).- If it is negative, there are two complex (non-real) roots.Since the discriminant is \(-31\), which is negative, there are two complex roots, which are conjugate pairs.

Step 4: Use the Quadratic Formula to Find Roots

The Quadratic Formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives the solutions to the quadratic equation:\( ax^2 + bx + c = 0 \).Substitute \( a = 4 \), \( b = -9 \), and \( c = 7 \) into the formula:\[ x = \frac{-(-9) \pm \sqrt{-31}}{2 \times 4} \]Simplify the expression:\[ x = \frac{9 \pm i\sqrt{31}}{8} \]The solutions are \( x = \frac{9 + i\sqrt{31}}{8} \) and \( x = \frac{9 - i\sqrt{31}}{8} \).

Key Concepts

Discriminant

The discriminant is a crucial component when working with quadratic equations. It helps us determine the nature of the roots without solving the entire equation. Given a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is defined as \( b^2 - 4ac \). This small expression tells a lot about the type and number of roots the equation has.

• If the discriminant is positive, the equation has two distinct real roots. This means the parabola intersects the x-axis at two points.
• If the discriminant equals zero, there is exactly one real root, often referred to as a repeated or double root. Here, the parabola touches the x-axis at a single point.
• If the discriminant is negative, as in our case with a value of \(-31\), the roots are not real numbers but complex conjugates. The parabola does not touch the x-axis at all. Instead, it hovers above or below depending on the orientation of the quadratic.

Understanding the discriminant helps predict the overall behavior of a quadratic function even before plotting or solving it completely.

Quadratic Formula

The Quadratic Formula is a renowned tool in mathematics, allowing us to find roots of any quadratic equation concisely. For any equation in the form \( ax^2 + bx + c = 0 \), it provides the solutions: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

This formula is incredibly valuable because:

• It works for all quadratic equations, irrespective of whether the roots are real or complex.
• It directly incorporates the discriminant \( b^2 - 4ac \), allowing one to apply the solutions and understand root nature simultaneously.
• It simplifies the often tedious root-finding process, requiring only straightforward arithmetic operations.

In the step-by-step solution of our exercise, using the values \( a = 4 \), \( b = -9 \), and \( c = 7 \), our application of the Quadratic Formula resulted in the complex roots. Simplifying gives us:\[ x = \frac{9 \pm i\sqrt{31}}{8} \] This highlights the quadratic formula's capability to seamlessly handle complex number solutions, showcasing its versatility beyond solving for just real numbers.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is negative, indicating that the roots are not real numbers but a pair of complex numbers. Complex roots usually appear in conjugate pairs, like \( a + bi \) and \( a - bi \). This reflects in their symmetrical nature on the complex plane.

In our solved example, the negative discriminant led us to discover complex roots:\[ x = \frac{9 + i\sqrt{31}}{8} \] and \[ x = \frac{9 - i\sqrt{31}}{8} \] Here, \( i \) represents the imaginary unit, equating to \( \sqrt{-1} \), and plays a pivotal role in structuring these roots. Complex roots, while not real, are significant in mathematical analysis and engineering fields.

• They appear in scenarios involving oscillations, waves, and alternate currents in electrical circuits.
• Understanding and working with complex roots can aid in better comprehension of polynomial behavior, even beyond quadratics.
• It introduces the fascinating world beyond the real number line, offering insight into higher-dimensional mathematics.

While they might seem abstract initially, complex roots are integral to various practical applications and help broaden mathematical horizons.