Chapter 1 Section 1.1

Chapter 1 Section 1.4Quadratic EquationsPrepared by Doron ShaharWarm-up: page 15A quadratic equation is an equation that can be written in the form _____________ where a, b, and c are constants and a 0.

The zero product property says that if , then either ________ or ________.

Prepared by Doron ShaharFOIL and Factoring

FirstOutsideInsideLastFOILFactorPrepared by Doron Shahar1.4.1 Solve by Factoring

Starting EquationFactorSolution

Zero Product PropertyPrepared by Doron Shahar1.4.2 Solve by Factoring

Starting EquationFactorSolution

Zero Product PropertyPrepared by Doron ShaharSolve by Factoring

Starting EquationFactorSolution

Zero Product Property

FOIL

Prepared by Doron ShaharIntro to Completing the square

Starting EquationTake square root of both sides of the equationSolution

Place on the right side of the equationPrepared by Doron ShaharIntro to Completing the square

Starting EquationInsert on right sideSolution

Take square root

Group like terms

Prepared by Doron ShaharGoal of Completing the squareThe goal of completing the squares is to get a quadratic equation into the following form:

Starting EquationInsert on right sideSolution

Take square root

Group like terms

eg.Prepared by Doron Shahar1.4.5 Completing the square

Starting Equation

Add 3 to both sidesMultiply both sides by 2Desired formThe goal of completing the squares is to get a quadratic equation into the following form:

Prepared by Doron ShaharExample: Completing the square

Starting Equation

Add 8 to both sidesMultiply both sides by 2Desired form

Prepared by Doron ShaharCompleting the square

Equation from previous slideInsert on right sideSolution

Take square root

Group like terms

Prepared by Doron Shahar1.4.2 General method of Completing the square

Starting EquationAdd 9 to both sidesAdd (8/2)2=16 to both sidesFactor left sidePrepared by Doron Shahar1.4.4 General method of Completing the square

Starting EquationDivide both sides by 3Add ((2/3)/2)2=1/9to both sidesFactor left sidePrepared by Doron Shahar1.4.3 General method of Completing the square

Starting EquationSubtract 22 from both sidesAdd (6/2)2=9to both sidesFactor left sidePrepared by Doron ShaharNo solutions in quadratic equationSolving by factoring works only if the equation has a solution.

Completing the square always works, and can be used to determine whether a quadratic equation has a solution.

Try solvingTake square root PROBLEMA! You CANNOT take the square root of a negative number. Therefore, the equation has no solution.Prepared by Doron ShaharGeneral method of Completing the square

Starting EquationSubtract c from both sidesAdd ((b/a)/2)2=(b/2a)2to both sidesFactor left sideDivide both sides by a

Prepared by Doron ShaharQuadratic Formula

If we solve for in the previous equation, ,we get an equation called the quadratic formula.

Quadratic FormulaStarting Equation

The quadratic formula gives us the solutions to every quadratic equation.

SolutionPrepared by Doron ShaharQuadratic Formula Song

Quadratic FormulaPlease sing along. Prepared by Doron ShaharUsing the quadratic formulaStarting Equation

Solution

Quadratic FormulaPlug 9 in for a, 6 for b, and 1 for c in the quadratic formula. Prepared by Doron ShaharSimplify your solution

Simplify

SolutionPrepared by Doron Shahar1.4.1 Using the quadratic formulaStarting Equation

Solution

Quadratic FormulaPlug 1 in for a, 9 for b, and 14 for c in the quadratic formula. 1Prepared by Doron ShaharSimplify your solution

Simplify

SolutionPrepared by Doron Shahar1.4.3 Using the quadratic formulaStarting Equation

Solution

Quadratic FormulaPlug 1 in for a, 6 for b, and 22 for c in the quadratic formula. 1+( )Prepared by Doron ShaharSimplify your solution

Simplify

You cannot take the square root of a negative number. Therefore, there is no solution.No SolutionPrepared by Doron ShaharDiscriminantEquationDiscriminant# of Solutions

Prepared by Doron ShaharCalculatorPut the Quadratic Formula program on your calculator. Instructions are in the back of the class notes.ORYou can come in to office hours to have me load the program onto your calculator.

Warning! The calculator will not always give you exact answers.Prepared by Doron ShaharSimplifying expressions with

Prepared by Doron ShaharQuadratic equations with DecimalsIf a quadratic equation has decimals, it is easiest to simply use the quadratic formula. If you want, you can multiply both sides of the equation by a power of 10 (i.e., 10, 100, 1000, etc) to get rid of the decimals. This can make it easier to simplify the answer if you are evaluating the quadratic formula without a calculator.

Prepared by Doron ShaharQuadratic equations with FractionsIf a quadratic equation has fractions (and does not factor), it is often easy to simply use the quadratic formula. If you want, you can multiply both sides of the equation by the least common denominator to get rid of the fractions. This can make it easier to simplify the answer.

If a quadratic equation has fractions (and factors), it is often easier to factor after having gotten rid of the fractions.

Prepared by Doron Shahar

Variables in the denominatorsIf an equation has variables in the denominator, it is NOT a quadratic equation. Such equations, however, can lead to linear equations. We treat such equations like those with fractions. That is, we multiply both sides of the equation by a common denominator to get rid of the variables in the dominators. Ideally, we should multiply by the least common denominator.

Example: If our problem has B +16 in the denominator of one term, and B in the denominator of another term, we multiply both sides of the equation by B(B+16). After the multiplication, the terms will have no variables in the denominators.

Prepared by Doron ShaharVariables in the denominators

Starting EquationMultiply both sides by B(B+16)

The problem is now in a form you can solve.Distribute the B(B+16) Prepared by Doron ShaharSystems of equationsThere are two methods for solving systems of equations: Substitution and Elimination. Both work by combining the equations into a single equation with one variable. And sometimes the resulting equation leads to a quadratic equation.

We will only review substitution, because elimination is not a common method when working with systems of equations that lead to quadratic equations.Prepared by Doron ShaharSubstitution

Starting Equations

Substitute B+16 for J in the second equationSolution for B Plug in 24 and 10 for B in the first equation to get the solution for JPrepared by Doron Shahar