/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Write a quadratic equation havin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 32

Write a quadratic equation having the given numbers as solutions. \(-5,\) only solution

Short Answer

Expert verified
The quadratic equation is \(x^2 + 10x + 25 = 0\).

Step by step solution

01

- Understand the Given Solution

Recognize that the given number \(-5\) is the only solution to the quadratic equation. This means it is a repeated root.
02

- Identify the Factored Form

A quadratic equation with a repeated root \(r\) can be expressed as \((x - r)^2 = 0\). Here, \(-5\) is the root.
03

- Substitute the Root

Substitute \(-5\) for \(r\) in the factored form: \((x - (-5))^2 = 0\). Simplify it to \((x + 5)^2 = 0\).
04

- Expand the Factored Form

Expand \((x + 5)^2 = 0\). This becomes \(x^2 + 10x + 25 = 0\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

repeated root
When solving quadratic equations, you might come across a term called a 'repeated root'. This happens when a quadratic equation has only one unique solution that appears twice. For example, if the root of the quadratic equation is \(x = -5\), it means \(x + 5\) appears twice in the factorized form of the equation. In mathematical terms, this is written as \((x + 5)^2 = 0\).

Here are some key points to remember about repeated roots:
  • A repeated root occurs when the quadratic equation touches the x-axis at exactly one point.
  • This usually happens in 'perfect square trinomials'.
  • The graph will have a vertex at the repeated root.

factored form
In many cases, especially if you have a repeated root, you will encounter equations in their 'factored form'. The factored form of a quadratic equation reveals information about its roots directly.

For an equation with a root r, it can be written as: \((x - r)^2 = 0\). Given that \(r = -5\), the factored form will be \((x + 5)^2 = 0\).

Let's break it down:
  • The equation \((x + 5)\) highlights there is a root at \(x = -5\).
  • Squaring it, \((x + 5)^2\) reinforces that this root is repeated.
This format is handy because once you recognize the root, it's straightforward to write it in a factored form.
expansion
Once you have the equation in factored form, it's crucial to understand how to 'expand' it back into the standard quadratic form. Expansion involves multiplying out the factors.

Let's use our example \((x + 5)^2 = 0\). To expand it:
  • First, write \((x + 5)(x + 5)\).
  • Then, use the distributive property (also known as FOIL in some textbooks): \(x^2 + 5x + 5x + 25\).
  • Combine like terms to get \(x^2 + 10x + 25\).

Therefore, \((x + 5)^2 = 0\) expands to \(x^2 + 10x + 25 = 0\). Expansion is essential because it converts the equation back into a familiar quadratic format, making it easier to solve and understand.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve. $$ 3 t^{2}-1=6 $$

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Simplify. \([10.3],[10.8]\) $$ \sqrt{-4} $$

Solve by completing the square. Show your work. $$ x^{2}+16 x+15=0 $$

For each equation under the given condition, (a) find \(k\) and (b) find the other solution. Write a quadratic equation with integer coefficients for which \(-\sqrt{2}\) is one solution.

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+9 x-25 $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.