/*! 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 46 Solve each equation by factoring... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 46

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 3 x^{2}-6 x+9=6 $$

Short Answer

Expert verified
The solution is \(x = 1\).

Step by step solution

01

Move all terms to one side

First, let's start by setting the equation in standard quadratic form. We need to move all terms to one side of the equation:\[3x^2 - 6x + 9 = 6\]Subtract 6 from both sides:\[3x^2 - 6x + 9 - 6 = 0\]This simplifies to:\[3x^2 - 6x + 3 = 0\]
02

Identify and simplify the quadratic equation

Look at the quadratic equation we have:\[3x^2 - 6x + 3 = 0\]Notice that all terms can be divided by 3 to simplify the equation:\[x^2 - 2x + 1 = 0\]
03

Factor the quadratic equation

Now we'll factor the quadratic equation. We look at:\[x^2 - 2x + 1 = 0\]The expression is a perfect square trinomial. It can be factored as:\[(x - 1)^2 = 0\]
04

Solve for the variable

We solve the factored equation:\[(x - 1)^2 = 0\]To solve, we take the square root of both sides to get:\[x - 1 = 0\]Then, add 1 to both sides:\[x = 1\]

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.

Factoring Quadratics
Understanding how to factor quadratics is crucial for solving certain quadratic equations. In such cases, we write the quadratic as a product of two binomials. This can make the process of finding solutions much simpler.
For a quadratic in standard form, such as \[ax^2 + bx + c = 0,\]our aim is to express it as \[(x - p)(x - q) = 0.\]This means that if we can find numbers \(p\) and \(q\) such that \(pq = c\) and \(p + q = b,\) then the quadratic can be factored.
  • The equation \(x^2 - 2x + 1 = 0\) is an example of a factorable quadratic.
  • By recognizing it as a perfect square trinomial, it mellows into \((x - 1)^2 = 0.\)
Factoring quadratics this way, especially recognizing patterns like perfect square trinomials, can be a powerful technique to efficiently solve quadratic equations.
Quadratic Formula
When factoring is complex or not feasible, the quadratic formula is our reliable friend. It provides a surefire method to find the roots of any quadratic equation. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula requires the coefficients \(a\), \(b\), and \(c\) from the quadratic equation in the standard form \(ax^2 + bx + c = 0\).
  • This formula helps when the quadratic doesn't factor neatly.
  • The term under the square root, \(b^2 - 4ac\), is called the discriminant.
  • The discriminant tells us about the nature of the roots: two distinct real roots, one real root, or no real roots.
The quadratic formula can solve any quadratic equation, making it a fundamental part of algebra.
Perfect Square Trinomials
A perfect square trinomial is a quadratic that can be represented as the square of a binomial. It has a particular structure, which allows for recognizing and factoring them efficiently. The general forms of perfect square trinomials are:
  • \((a + b)^2 = a^2 + 2ab + b^2,\) and
  • \((a - b)^2 = a^2 - 2ab + b^2.\)
Identifying perfect square trinomials allows for immediate factoring, simplifying the solving process. In our equation \(x^2 - 2x + 1 = 0,\) we spotted that it matches the pattern:
\[(x - 1)^2 = x^2 - 2x + 1.\]Recognizing such a pattern:
  • Saves time when solving equations.
  • Prepares students to solve more complex problems by breaking them down into simple parts.
This understanding helps students work through and solve equations efficiently, just like in our example.

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

i. Show that the general linear equation \(a x+b y=c\) with \(b \neq 0\) can be written as \(y=-\frac{a}{b} x+\frac{c}{b}\) which is the equation of a line in slope-intercept form. ii. Show that the general linear equation \(a x+b y=c\) with \(b=0\) but \(a \neq 0\) can be written as \(x=\frac{c}{a}\), which is the equation of a vertical line. [Note: Since these steps are reversible, parts (i) and (ii) together show that the general linear equation \(a x+b y=c\) (for \(a\) and \(b\) not both zero) includes vertical and nonvertical lines.]

$$ \text { How do the graphs of } f(x) \text { and } f(x)+10 \text { differ? } $$

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{3}{x} $$

\(77-78 .\) GENERAL: Impact Time of a Projectile If an object is thrown upward so that its height (in feet) above the ground \(t\) seconds after it is thrown is given by the function \(h(t)\) below, find when the object hits the ground. That is, find the positive value of \(t\) such that \(h(t)=0\). Give the answer correct to two decimal places. [Hint: Enter the function in terms of \(x\) rather than t. Use the ZERO operation, or TRACE and ZOOM IN, or similar operations.] $$ h(t)=-16 t^{2}+45 t+5 $$

SOCIAL SCIENCE: Immigration The percentage of immigrants in the United States has changed since World War I as shown in the following table. \begin{tabular}{llllllllll} \hline Decade & 1930 & 1940 & 1950 & 1960 & 1970 & 1980 & 1990 & 2000 & 2010 \\\ \hline Immigrant Percentage & \(11.7\) & \(8.9\) & 7 & \(5.6\) & \(4.9\) & \(6.2\) & \(7.9\) & \(11.3\) & \(12.5\) \\ \hline \end{tabular} a. Number the data columns with \(x\) -values \(0-8\) (so that \(x\) stands for decades since 1930 ) and use 0 quadratic regression to fit a parabola to the data. State the regression function. [Hint: See Example 10.] b. Use your curve to estimate the percentage in \(2016 .\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

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.