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Chapter 5: Problem 37

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

Short Answer

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

Step by step solution

01

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.
02

Substitute Roots into Factored Form

Given the roots \(1\) and \(0\), we substitute them into the factored form of a quadratic equation: \((x - 1)(x - 0) = 0\).
03

Expand the Factored Form

Next, expand the expression \((x - 1)(x - 0)\) to obtain the equation in standard form. The expansion is as follows: \(x(x - 1) = x^2 - x = 0\).
04

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 \(x^2 - x = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Write a quadratic equation for which the sum of the roots is equal to the product of the roots.

Write a quadratic equation with integer coefficients for each pair of roots. \(4,7\)

In \(9-17,\) graph each system and determine the common solution from the graph. $$ \begin{array}{l}{y=-x^{2}+6 x-1} \\ {y=x+3}\end{array} $$

The difference in the lengths of the sides of two squares is 1 meter. The difference in the areas of the squares is 13 square meters. What are the lengths of the sides of the squares?

a. On the same set of axes, sketch the graphs of \(y=x^{2}+5\) and \(y=2 x\) b. Does the system of equations \(y=x^{2}+5\) and \(y=2 x\) have a common solution in the set of real numbers? Justify your answer. c. Does the system of equations \(y=x^{2}+5\) and \(y=2 x\) have a common solution in the set of complex numbers? If so, find the solution.
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