Algebra I – Quadratic Equations - Quick Guides | Mathematics

Preview (10 of 33 Pages)

100%

Algebra I – Quadratic Equations - Page 1 preview image

Loading page ...

Study 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:

Algebra I – Quadratic Equations - Page 2 preview image

Loading page ...

Study 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

Algebra I – Quadratic Equations - Page 3 preview image

Loading page ...

Study 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

Algebra I – Quadratic Equations - Page 4 preview image

Loading page ...

Study 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)

Algebra I – Quadratic Equations - Page 5 preview image

Loading page ...

Study 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:

Algebra I – Quadratic Equations - Page 6 preview image

Loading page ...

Study 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]

Algebra I – Quadratic Equations - Page 7 preview image

Loading page ...

Study 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):

Algebra I – Quadratic Equations - Page 8 preview image

Loading page ...

Study 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,

Algebra I – Quadratic Equations - Page 9 preview image

Loading page ...

Study 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:

Algebra I – Quadratic Equations - Page 10 preview image

Loading page ...

Study 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!

Report

Algebra I – Quadratic Equations

Study Now!

Document Details

Related Documents

Algebra I – Word Problems

Algebra I – Terminology Sets and Expressions

Algebra I – Signed Numbers Fractions and Percents

Algebra I – Roots and Radicals

Algebra I – Preliminaries and Basic Operations

Algebra I – Monomials Polynomials and Factoring

Algebra I – Inequalities Graphing and Absolute Value

Algebra I – Functions and Variations

Algebra I – Equations with Two Variables

Algebra I – Equations Ratios and Proportions

Company

Explore

Study Tools