Problem 110 - Algebra and Trigonometry Real Mathematics, Real People

Question

Given that the solutions of a quadratic equation are \(x=(-b \pm \sqrt{b^{2}-4 a c}) /(2 a),\) show that the product of the solutions is \(P=c / a\).

Step-by-Step Solution

Answer

The product of the solutions of the quadratic equation is \( P = \frac{c}{a} \)

Step 1: Identify the two solutions to the quadratic equation

From the quadratic equation \( ax^2 + bx + c = 0 \), the solutions are given by \( x = \frac{-b + \sqrt{b^{2}-4ac}}{2a} \) and \( x = \frac{-b - \sqrt{b^{2}-4ac}}{2a} \). Let's call these solutions x1 and x2 respectively.

Step 2: Compute the product of the solutions

Multiply x1 and x2 together, which gives \( P = x1 * x2 = \frac{-b + \sqrt{b^{2}-4ac}}{2a} * \frac{-b - \sqrt{b^{2}-4ac}}{2a} \).

Step 3: Simplify the equation

The product \( P = (\frac{-b}{2a} + \frac{\sqrt{b^{2}-4ac}}{2a}) * (\frac{-b}{2a} - \frac{\sqrt{b^{2}-4ac}}{2a}) \) simplifies to \( P = \frac{b^2}{4a^2} - \frac{b^2 - 4ac}{4a^2} \).

Step 4: Further simplify the equation

The equation \( P = \frac{4ac}{4a^2} = \frac{c}{a} \), and thus, the product of the solutions of the quadratic equation is indeed \( P = \frac{c}{a} \).

Key Concepts

Solutions of Quadratic Equation

Solving a quadratic equation can initially seem daunting, but it's based on a systematic approach using a specific formula. A quadratic equation is generally expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable.
The solutions for \( x \) in a quadratic equation can be determined using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula yields two solutions, often denoted as \( x_1 \) and \( x_2 \). These represent the points where the parabola, defined by the quadratic equation, intersects the x-axis.
Understanding this concept is crucial for solving any quadratic equation regardless of its parameters. It's essential to use the signs correctly — the plus-minus symbol (\( \pm \)) indicates that there are two possible solutions.

Product of Solutions

An interesting property of quadratic equations is the relationship between its roots or solutions. For a quadratic equation of the standard form \( ax^2 + bx + c = 0 \), the product of the solutions can always be found using the relationship:

• \( P = \frac{c}{a} \)

This result comes from multiplying the two solutions \( x_1 \) and \( x_2 \). Mathematically, this is shown as:
\( P = x_1 \cdot x_2 \).
When expanded, these solutions take the form of complex expressions involving the quadratic formula. However, their internal mechanics involving algebraic properties simplify to \( P = \frac{c}{a} \). This formula provides a quick and easy way to determine the product without explicitly solving the equations.
This property holds true for all quadratic equations, making it a valuable tool in algebra.

Quadratic Formula

The quadratic formula is a cornerstone of algebra, offering a reliable method to solve any quadratic equation. The formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

It directly springs from the process of completing the square in the quadratic equation \( ax^2 + bx + c = 0 \).
Key components of the formula include:

• \(-b\), the opposite of the linear coefficient.
• \(\sqrt{b^2 - 4ac}\), known as the discriminant, determining the nature of the roots.
• \(2a\) in the denominator, normalizing the equation based on the leading coefficient.

The formula not only provides the solutions \( x_1 \) and \( x_2 \), but also allows understanding of the characteristics of the roots. Depending on the value of the discriminant, the roots can be real and distinct, real and repeated, or complex.
Mastering the quadratic formula is essential for anyone looking to delve deeper into algebra or any field relying on mathematical problem-solving.