/*! 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 7 Solve each equation by factoring... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 7

Solve each equation by factoring. $$a^{2}+3 a=-2$$

Short Answer

Expert verified
The solutions are $$a = -1$$ and $$a = -2$$.

Step by step solution

01

Set the Equation to Zero

Rewrite the equation so that all terms are on one side, resulting in a zero on the other side. $$a^2 + 3a + 2 = 0$$
02

Factor the Quadratic Equation

Identify two numbers that multiply to get the constant term (2) and add to give the coefficient of the linear term (3). These numbers are 1 and 2. So, the equation can be factored as: $$a^2 + 3a + 2 = (a + 1)(a + 2) = 0$$
03

Solve for the Variable

Set each factor equal to zero and solve for the variable:$$a + 1 = 0$$$$a = -1$$$$a + 2 = 0$$$$a = -2$$

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
Factoring is a method used to break down complex expressions into simpler ones that, when multiplied together, give the original expression.
In the context of quadratic equations, we focus on expressions of the form \(ax^2 + bx + c\).
The goal is to express them as the product of two binomials.
For example, in the equation \(a^2 + 3a + 2 = 0\), we need to find two numbers that both add up to 3 and multiply to 2.
Those two numbers are 1 and 2. This allows us to factor the quadratic as \((a + 1)(a + 2) = 0\).
Factoring simplifies solving equations by breaking them down into solvable parts.
quadratic equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\).
They graph as parabolas and have various methods for finding their solutions or roots.
One such method is factoring, which transforms the quadratic into products of binomials.
Quadratic equations can have two solutions, one solution, or no real solution depending on the discriminant value \( b^2 - 4ac \).
In our example, the quadratic equation is arranged as \(a^2 + 3a + 2 = 0\).
Factoring it resulted in \((a + 1)(a + 2) = 0\) from which we solve for the variables.
roots of equations
The roots of a quadratic equation are the values of the variable that satisfy the equation.
For the equation \((a + 1)(a + 2) = 0\), setting each binomial equal to zero, we solve for the roots.
\((a + 1) = 0\) gives us \(a = -1\), and \((a + 2) = 0\) gives us \(a = -2\).
These roots, \(-1 \) and \(-2\), are the solutions to the original quadratic equation.
The roots represent the points where the parabola, defined by the quadratic equation, intersects the x-axis.
They are critical for graphing and understanding the behavior of the function.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols.
In algebra, we learn to solve equations and find unknown values.
The quadratic equation \(a^2 + 3a + 2 = 0\) is a perfect example where we use algebra to find solutions.
We rearrange terms, factor the quadratic, and apply rules to solve for the variable.
Techniques such as factoring are foundational in algebra as they simplify complex expressions and make equations solvable.
Mastering these basics is essential for understanding more advanced mathematical concepts.

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 each equation. $$(x-3)^{2}=x^{2}-9$$

Find an exact solution to each problem. If the solution is irrational, then find an approximate solution also. Speed of an Electric Car An experimental electric-solar car completed a 1000 -mi race in 35 hr. For the 600 mi traveled during daylight the car averaged 20 mph more than it did for the 400 mi traveled at night. What was the average speed of the car during the daytime?

Solve each equation. $$\frac{2}{x-1}-\frac{3}{x+2}=\frac{8-x}{x^{2}+x-2}$$

Write an inequality of the form \(|x-a| < k\) or of the form \(|x-a| > k\) so that the inequality has the given solution set. HINT: \(|x-a| < k\) means that \(x\) is less than \(k\) units from \(a\) and \(|x-a|>k\) means that \(x\) is more than \(k\) units from \(a\) on the number line. $$(4,8)$$

Solve each equation. $$-\frac{1}{6}(x+3)=\frac{1}{4}(3-x)$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.