Problem 37 - Algebra 2 and Trigonometry

Question

Write a quadratic equation with integer coefficients for each pair of roots. \(1,0\)

Step-by-Step Solution

Answer

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

Step 1: Understanding Quadratic Equation and Roots

A quadratic equation in standard form is represented as \(ax^2 + bx + c = 0\). The roots of the equation are values of \(x\) that satisfy the equation. Given the roots, we can form a quadratic equation by using the factored form \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots.

Step 2: Substitute Roots into Factored Form

Given the roots \(1\) and \(0\), we substitute them into the factored form of a quadratic equation: \((x - 1)(x - 0) = 0\).

Step 3: Expand the Factored Form

Next, expand the expression \((x - 1)(x - 0)\) to obtain the equation in standard form. The expansion is as follows: \(x(x - 1) = x^2 - x = 0\).

Step 4: Write the Quadratic Equation

The expanded form \(x^2 - x = 0\) is already a quadratic equation with integer coefficients. Therefore, the quadratic equation with roots \(1\) and \(0\) is \(x^2 - x = 0\).

Key Concepts

Factored Form

One of the easiest ways to create a quadratic equation when you know its roots is by using the factored form. The factored form of a quadratic equation is written as \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are the roots. This represents the reverse process of solving a quadratic equation by factoring.

• "Factored form" essentially means expressing a polynomial as a product of its linear factors.
• It's a helpful form because it shows the roots (or solutions) of the equation at a glance.

For example, if you have roots \(1\) and \(0\), the factored form becomes \((x - 1)(x - 0)\).
This makes it simple to see that when \(x = 1\) or \(x = 0\), the entire equation equals zero.

Integer Coefficients

Integer coefficients are a specific requirement for the quadratic equations in many exercises. This means that all the constants and coefficients in the quadratic equation should be whole numbers.

• In a quadratic equation, integer coefficients make calculations straightforward and limit the complexity.
• They help ensure results that are easy to interpret, especially in a basic algebra context.

Using the example from the problem, we see that the quadratic equation \(x^2 - x = 0\) has integer coefficients. Here, the coefficient of \(x^2\) is 1, for \(x\) it is -1, and the constant term is 0. All these terms meet the requirement of being integers.

Roots of a Quadratic Equation

The roots (or solutions) of a quadratic equation are the values of \(x\) that satisfy the equation, making it equal to zero. Finding the roots means understanding where the equation crosses the x-axis on a graph.

• Quadratic equations can have two real roots, one real root, or complex roots.
• The roots can be found by factoring the equation, using the quadratic formula, or completing the square.

Given the roots \(1\) and \(0\) in our exercise, directly inserting them into a factored form provides \((x - 1)(x - 0) = 0\).
This equation immediately tells us the roots, hence making our job of back-calculating the original equation straightforward. This confirms why understanding the roots is key to both forming and solving quadratic equations efficiently.