Problem 82 - Elementary Algebra

Question

Find a quadratic equation with integer coefficients, given the following solutions. $$ -5,3 $$

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + 2x - 15 = 0\).

Step 1: Understand the Problem

We need to find a quadratic equation whose solutions are \(-5\) and \(3\). Quadratic equations in standard form are expressed as \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are integers.

Step 2: Use the Factored Form

When a quadratic equation has known roots, it can be written as \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots. For our solutions, this gives us \((x - (-5))(x - 3) = 0\) or \((x + 5)(x - 3) = 0\).

Step 3: Expand the Expression

Expand the factored form \((x + 5)(x - 3)\) using the distributive property: \((x + 5)(x - 3) = x(x - 3) + 5(x - 3) = x^2 - 3x + 5x - 15\).

Step 4: Simplify the Expression

Combine like terms in the expanded expression: \(x^2 + 2x - 15 = 0\). This is the quadratic equation in standard form with integer coefficients.

Key Concepts

Integer Coefficients

In the context of a quadratic equation, integer coefficients refer to the values of \(a\), \(b\), and \(c\) in the expression \(ax^2 + bx + c = 0\), where all these numbers are whole numbers. Whole numbers include all positive and negative numbers, including zero, without any fractional or decimal parts. Using integer coefficients is critical because it often simplifies the equation and is essential in many practical scenarios, especially in textbook problems or computer algorithms, where precise values are necessary. In our example, we've found the quadratic equation to be \(x^2 + 2x - 15 = 0\). Here

• \(a = 1\)
• \(b = 2\)
• \(c = -15\)

All of these coefficients are integers, fulfilling the requirement laid out by the problem.

Factored Form

The factored form of a quadratic equation gives us a way to express the equation in terms of its roots, or solutions. When we know these roots, we can write the quadratic equation as \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots. In this problem, the roots are given as \(-5\) and \(3\). Thus, we write the equation in factored form as \((x + 5)(x - 3) = 0\). Factored form is immensely useful:

• It provides a direct way to find the roots of the equation just by solving the factors.
• It can be easily expanded to find the standard form of the equation.

This form helps us understand the structure of the equation before we transform it into its expanded form, also known as standard form.

Distributive Property

The distributive property is a fundamental mathematical principle used to expand expressions, especially helpful when working with factored forms. According to the distributive property, \(a(b + c) = ab + ac\). In the case of quadratic equations, this property allows us to take a factored equation like \((x + 5)(x - 3)\) and expand it into the polynomial form:

• First, multiply \(x\) by each term in \((x - 3)\) to get \(x^2 - 3x\).
• Next, take \(5\) and multiply by each term in \((x - 3)\) to get \(5x - 15\).
• Finally, add all the terms together to get \(x^2 - 3x + 5x - 15\).

After simplifying, this results in \(x^2 + 2x - 15\), which is the quadratic equation in its standard form.

Standard Form

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c = 0\). This form is useful because it presents the equation in a way that makes it easy to recognize its coefficients \(a\), \(b\), and \(c\). Let's look at the equation we derived: \(x^2 + 2x - 15 = 0\). Here the coefficients are:

• \(a = 1\)
• \(b = 2\)
• \(c = -15\)

The standard form is versatile for solving quadratic equations using various methods, such as factoring, completing the square, or applying the quadratic formula. Understanding the standard form is key to moving between different formats of equations and leveraging each format's advantages for solving and analyzing quadratic equations efficiently.