Problem 91 - Elementary Algebra

Question

Find a quadratic equation with integer coefficients, given the following solutions. $$ -1 / 5,-2 / 3 $$

Step-by-Step Solution

Answer

The quadratic equation is \(15x^2 + 13x + 2 = 0\).

Step 1: Understanding the Problem

We are asked to find a quadratic equation when given its solutions, namely \(-\frac{1}{5}\) and \(-\frac{2}{3}\). The quadratic equation with these roots will have the form \(a(x - r_1)(x - r_2) = 0\) where \(r_1\) and \(r_2\) are the roots.

Step 2: Writing the Root Factor Form

The factor form of the quadratic equation using the roots \(-\frac{1}{5}\) and \(-\frac{2}{3}\) is: \[(x + \frac{1}{5})(x + \frac{2}{3}) = 0\]This step represents the equation where each root provides a linear factor.

Step 3: Eliminating Fractions

To eliminate fractions, multiply each term of the equation by the least common multiple (LCM) of the denominators, which is 15. Multiplying through gives: \[15(x + \frac{1}{5})(x + \frac{2}{3}) = 0\] This simplifies to:\[(5x + 1)(3x + 2) = 0\] by expanding each factor by 15 and then simplifying.

Step 4: Expanding the Expression

Expand the expression \[(5x + 1)(3x + 2)\]:\[ 5x \cdot 3x + 5x \cdot 2 + 1 \cdot 3x + 1 \cdot 2 = 0\]Which results in:\[ 15x^2 + 10x + 3x + 2 = 0\]Combine like terms to get:\[ 15x^2 + 13x + 2 = 0\]

Step 5: Verifying Integer Coefficients

Check that the equation \(15x^2 + 13x + 2 = 0\) has integer coefficients, which it does. Therefore, it satisfies the condition of the problem to have integer coefficients.

Key Concepts

Integer Coefficients

When dealing with quadratic equations, having integer coefficients means that all the coefficients in the equation are whole numbers. In the context of a quadratic equation, which typically takes the form \( ax^2 + bx + c = 0 \), the integers \( a \), \( b \), and \( c \) are required.

This is an important aspect because working with integers simplifies calculations and avoids errors related to fractions or decimals. For example, when converting a quadratic equation with fractional roots to a version with integer coefficients, it ensures that the final equation is easier to handle.

Here's how to ensure integer coefficients:

• Identify the roots of the equation you have.
• Formulate the equation using root-based factors.
• Eliminate any fractions by multiplying through by the least common multiple of the denominators.
• Finally, expand and simplify the equation to confirm all coefficients are integers.

In our solution, we went from fraction-based expressions to ensure that the coefficients for \( 15x^2 + 13x + 2 = 0 \) are integers.

Quadratic Formula

The quadratic formula is a universal method in algebra used for solving quadratic equations. It's particularly useful when factoring is not straightforward. However, in problems like the one we're addressing, where roots are given, we use the idea behind the quadratic formula in reverse.

The typical formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Given the roots, we express the quadratic equation as a product of linear factors, \((x - r_1)(x - r_2)\). While this may bypass the direct use of the quadratic formula, knowing it provides insight into why equations are structured as they are.

In our example, this reverse process is done by first acknowledging the roots expressed in fractions, \(-\frac{1}{5}\) and \(-\frac{2}{3}\), then construct the equation from these roots leveraging integer coefficients. The quadratic formula validates that these roots satisfy the simplified equation.

Roots of Equations

The roots of a quadratic equation are solutions for which the equation equals zero. In a quadratic like \( ax^2 + bx + c = 0 \), roots \( r_1 \) and \( r_2 \) are the values of \( x \) where the equation equates to zero.

When you know the roots already, constructing the equation involves:

• Converting each root into a factor form \((x - r)\).
• Then multiplying these factors out to form the quadratic equation.

The complication in our problem was that roots were fractions. To represent them precisely, we transformed them into factors \((x + \frac{1}{5})(x + \frac{2}{3})\) and accordingly eliminated fractions to derive a clear and workable quadratic equation.

Understanding roots this way not only helps in solving problems backward from known solutions but also gives deeper insight into relationships within quadratic equations. After simplifying to integer coefficients, roots validate our resulting equation \(15x^2 + 13x + 2 = 0\).