CramX Logo
RegisterLog in
Algebra I – Quadratic Equations - Document preview page 1

Algebra I – Quadratic Equations - Page 1

Document preview content for Algebra I – Quadratic Equations

Algebra I – Quadratic Equations

This document provides study materials related to Algebra I – Quadratic Equations. It may include explanations, summarized notes, examples, or practice questions designed to help students understand key concepts and review important topics covered in their coursework.

Students studying Mathematics or related courses can use this material as a reference when preparing for assignments, exams, or classroom discussions. Resources on CramX may include study notes, exam guides, solutions, lecture summaries, and other academic learning materials.

Nivaldo
Contributor
0.0
0
2 months ago
Preview (10 of 33 Pages)
100%
Log in to unlock
Page 1 of 33
Algebra I – Quadratic Equations - Page 1 preview imageStudy GuideAlgebra I–QuadraƟc EquaƟons1. Solving QuadraƟc EquaƟonsWhat Is a Quadratic Equation?Aquadratic equationis an equation that can be written in this standard form:Here:•(a),(b), and(c)are numbers,•(x)is the variable,•and(a≠0)(because if(a = 0), the equation would not be quadratic anymore).Quadratic equations always include an(x2)term. That squared term is what makes them differentfrom linear equations.Ways to Solve a Quadratic EquationThere arethree main methodsfor solving quadratic equations:1.Factoring2.Using the quadratic formula3.Completing the squareIn this section, we’ll focus onfactoring.Solving by FactoringFactoring is often the quickest method—when the equation can be factored easily.Steps for FactoringFollow these steps carefully:
Page 2 of 33
Algebra I – Quadratic Equations - Page 2 preview imageStudy Guide1.Move all terms to one sideof the equation so that the other side equals 0.2.Factorthe expression.3.Set each factor equal to zero.4.Solvethe resulting equations.5.Check your answersby plugging them back into the original equation.Let’s see how this works in an example.Example 1Solve:[x2-6x = 16]Step 1:Move all terms to one sideSubtract 16 from both sides so the equation equals 0:Step 2:FactorNow factor the quadratic expression:Step 3:Set each factor equal to zeroStep 4:Solve each equation
Page 3 of 33
Algebra I – Quadratic Equations - Page 3 preview imageStudy GuideSo we havetwo possible solutions:Step 5:Check Your AnswersAlways substitute your answers back into the original equation to make sure they work.Original equation:Check(x = 8)Check(x =-2)Final AnswerThe solutions are:Example 2Solve:[y2=-6y-5]Step 1:Move all terms to one side
Page 4 of 33
Algebra I – Quadratic Equations - Page 4 preview imageStudy GuideTo factor, we need the equation to equal 0.Add (6y) and (5) to both sides:Step 2:FactorNow factor the quadratic expression:Step 3:Set each factor equal to zeroStep 4:SolveSo the solutions are:Step 5:Check Your AnswersSubstitute each value into the original equation:Original equation:Check(y =-5)
Page 5 of 33
Algebra I – Quadratic Equations - Page 5 preview imageStudy GuideLeft side:Right side:25 = 25 (Correct)Check(y =-1)Left side:Right side:1 = 1 (Correct)Both values work!Example 3Solve:[x2-16 = 0]Step 1:FactorThis is adifference of squares.So we write:
Page 6 of 33
Algebra I – Quadratic Equations - Page 6 preview imageStudy GuideStep 2:Set each factor equal to zeroStep 3:SolveSo the solutions are:Step 4:CheckSubstitute into the original equation(x2-16 = 0).For(x =-4):For(x = 4):Both solutions satisfy the equation.Example 4Solve:[x2+ 6x = 0]
Page 7 of 33
Algebra I – Quadratic Equations - Page 7 preview imageStudy GuideStep 1:FactorHere, there isno constant term. This is an incomplete quadratic.Factor out the greatest common factor (GCF), which is(x):Step 2:Set each factor equal to zeroorStep 3:SolveSo the solutions are:Step 4:CheckFor(x = 0):For(x =-6):
Page 8 of 33
Algebra I – Quadratic Equations - Page 8 preview imageStudy GuideBoth solutions are correct.Example 5Solve:Step 1:Move Everything to One SideTo solve by factoring, the equation must equal0.Subtract (x2+ 6x-5)from both sides and combine like terms:Now simplify:Step 2:FactorThis is aperfect square trinomial:or written more simply,
Page 9 of 33
Algebra I – Quadratic Equations - Page 9 preview imageStudy GuideStep 3:SolveThis equation hasone solution.(It is sometimes called a repeated or double root.)Step 4:Check Your AnswerSubstitute(x = 2)into the original equation:Simplify both sides:Left side:Right side:The Quadratic FormulaFactoring works wellwhen the quadratic can be factored easily.However, many quadratic equationscannotbe solved by factoring. This is especially true when thesolutions arenot rational numbers.In those cases, we use thequadratic formula:
Page 10 of 33
Algebra I – Quadratic Equations - Page 10 preview imageStudy GuideUnderstanding the FormulaThe quadratic formula is used when the equation is written instandard form:Where:•(a)is the number in front of(x2)•(b)is the number in front of(x)•(c)is the constant (the number without a variable)The DiscriminantInside the square root is an important expression:This part is called thediscriminant.The discriminant tells you how many real solutions the equation has.There arethree possible cases:1.If(b2-4ac > 0)→The equation hastwo different real roots.2.If(b2-4ac = 0)→The equation hasone real root(a repeated root).3.If(b2-4ac < 0)→The equation hasno real roots.Example 6Solve for (x):
Preview Mode

This document has 33 pages. Sign in to access the full document!

Study Now!

X-Copilot AI
Unlimited Access
Secure Payment
Instant Access
24/7 Support
Document Chat

Document Details

University
Texas A&M University
Course
Linear Algebra
Subject
Mathematics

Related Documents

View all
Quick Guides

Algebra I – Word Problems

Quick Guides

Algebra I – Terminology Sets and Expressions

Quick Guides

Algebra I – Signed Numbers Fractions and Percents

Quick Guides

Algebra I – Roots and Radicals

Quick Guides

Algebra I – Preliminaries and Basic Operations

Quick Guides

Algebra I – Monomials Polynomials and Factoring

Quick Guides

Algebra I – Inequalities Graphing and Absolute Value

Quick Guides

Algebra I – Functions and Variations

Quick Guides

Algebra I – Equations with Two Variables

Quick Guides

Algebra I – Equations Ratios and Proportions