Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 241.2 Quadratic EquationsChapter 1 Section 2: Quadratic EquationsIn this section, we will… Solve quadratic equations by factoring Solve quadratic equations by extraction of roots Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula (real solutions only) Use the discriminant to determine the nature of the real solutions of a given quadratic equation Solve applications involving quadratic equations1.2 Quadratic Equationsstandard form2A is any equation that can be written in the form 0where , , and are real numbers and 0.ax bx ca b c a+ + =�We will solve quadratic equations by:• factoring• extraction of roots• completing the square• using the quadratic formulapossible outcomes:quadratic equation1.2 Quadratic Equations: Solving Quadratic Equations by FactoringSolving Quadratic Equations by Factoring:1. Write the quadratic equation in standard form (i.e. set the equation equal to 0)2. Factor completely3. Use the Zero-Product Rule 4. Solve the resulting linear equations5. Check your potential solution(s)We will solve by factoring when, once set equal to zero, the result is factorable.If and are real numbers, then0 0 or b 0a bab a= � = =1.2 Quadratic Equations: Solving Quadratic Equations by FacoringExamples: Solve each equation by factoring. Check your result(s).22 50 0x - =225 16 40x x+ =checkscheck1.2 Quadratic Equations: Solving Quadratic Equations by FactoringExample: Solve the equation by factoring. Check your result(s).22(2 4 ) 3 0y y- + = checks1.2 Quadratic Equations: Solving Quadratic Equations by Extraction of RootsSolving Quadratic Equations by Extracting Roots:1. Write the quadratic equation in the form2. Take the square root of both sides of the equationIf c < 0, there will be no real solutionsIf c = 0, there will be one real solutionIf c > 0, there will be two real solutions 3. Check your potential solution(s)We will solve by extracting rootswhen our equation has the form2x c=2x c=1.2 Quadratic Equations: Solving Quadratic Equations by Extraction of RootsExamples: Solve each equation by extracting the roots (a.k.a. the square root method). Check your result(s).212x =( )23 2 4x - =checks checks1.2 Quadratic Equations: Solving Quadratic Equations by Completing the SquareCompleting the Square: Recall our perfect squares from Review Section 4 example:We will be reversing this process and filling in the blanks. examples:Special Product FormulasPerfect Squares( )22 22x a x ax a+ = + +( )22 22x a x ax a- = - +( )24x + =Take half of the x-term coefficientIt goes here( )22 22x a x ax a+ = + +( )228 ___ ___x x x+ + = +Now square that resultIt goes here22x x-26x x-2x x+1.2 Quadratic Equations: Solving Quadratic Equations by Completing the SquareSolving Quadratic Equations by Completing the Square: 1. Write the quadratic equation in the form2. Make sure a = 1… if it is not, divide all terms by a3. Complete the square•Find half of b•Square the result4. Solve the resulting equation by extraction of roots5. Check your potential solution(s)We can solve any quadratic eq. this way!2ax bx c+ =1.2 Quadratic Equations: Solving Quadratic Equations by Completing the SquareExamples: Solve the equation by completing the square. Check your result(s).28 7 0x x- + =checks1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square24 7 0x x+ + =checksExamples: Solve the equation by completing the square. Check your result(s).1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square210 7 0x x+ - =checksExamples: Solve the equation by completing the square. Check your result(s).1.2 Quadratic Equations: Solving Quadratic Equations by Completing the Square28 9 6x x= -checksExamples: Solve the equation by completing the square. Check your result(s).1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic FormulaSolving Quadratic Equations by Using the Quadratic Formula: 1. Write the quadratic equation in the form2. Use the quadratic formula to find the solution(s)3. Check your potential solution(s)We can solve any quadratic eq. this way!20 , 0ax bx c a+ + = �242b b acxa- � -=We can use the discriminant to determine the nature of the real solutions to our given quadratic equation.If the discriminant of is:• negative, then there are no real solutions• zero, then there is one real solution• positive, then there are two different real solutions24b ac-1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic Formula26 5x x- =Examples: Solve the equation by using the quadratic formula:Check your result(s).1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic Formula22 4 1 0x x+ + =Examples: Solve the equation by using the quadratic formula:Check your result(s).1.2 Quadratic Equations: Solving Quad. Equations by Using the Quadratic Formula24 6 9x x= -Examples: Solve the equation by using the quadratic formula:Check your result(s).24 7 0x x+ + =Examples: Use the discriminant to determine the nature of the real solutions of the following quadratic equations.225 4 20x x+ =22 3 7 0x x- - =1.2 Quadratic Equations: Using the Discriminant to Determine the Nature of SolutionsSummary of Techniques:We can now solve quadratic equations by:1. Factoringplace in formfactor and use zero-product rule2. Extracting the rootsplace in formtake the square root of both sides3. Completing the squareplace in formcomplete the squaretake the square root of both sides4. Using the quadratic formulaplace in the formuse the quadratic formula1.2 Quadratic Equations2x c=20ax bx c+ + =20ax bx c+ + =2ax bx c+ =242b b acxa- � -=Take half of the x-term coefficientIt goes here( )22 22x a x ax a+ = + +Now square that resultIt goes hereCan use to solve any quad. eq.Can only solve by factoring if this is factorableCan only extract roots if there is no x-term1.2 Quadratic Equations: Solving Applications Involving Quadratic EquationsHow to Solve a Word Problem:Step 1: Read the problem until you understand it. • What are we asked to find? • What information is given?• What vocabulary is being used?Step 2: Assign a variable to represent what you are looking for. Express any remaining