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Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable Vocabulary: Factor: breaking up the equation into binomials Solution, zeros, x – intercept : solving the equation for x. Quadratic Equation standard form is ax2 + bx + c (a, b, c are real numbers) Part 1) Factor using GCF Method 1) 6 and 12 2) 6 and 15 3) 8x2 + 2x 4) 3x3y – 6x2y2 – 3xy3 Part 2 – All Trinomials – Factor using KrissKross, state the x-intercepts 1) x2 + 5x + 4 2) x2 – 8x + 12 3) 6x2 + 5xy – 4y2 4) 2x2 + 7xy – 4y2 5) x2 + 2x + 5 6) 6x4 + 11x2 – 10 Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable Part 3 – Find the solution using square root What is different with these equations compared to part 2 equations? Difference of Square (DOS) u2 - v2 = (u + v)(u - v) 1) r2 – 9 2) x2 – 25 3) 100 – m2 4) 9x2 – 1 5) 49x2 – 100 6) 3d2 – 48 7) 11x2 – 10 8) 2x2 – 15 Part 4 – Factor by Quadratic formula What is different with these equations compared to part 2 equations? Quadratic Formula: Discriminant b b 2 4ac x 2a when given ax2 + bx + c = 0 a, b, c are the coefficients Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have. Examples: 1: 3x2 + 8x - 14 = 0 a) a = b= c= b) Disc. = c) solution = d) factors: 2: 4x2 + 20x + 25 = 0 a) a= b= c= b) Disc. = c) zeros = d) factors: 3) x2 - 9x = 18 4) 25x2 + 9 = 30x Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable Recall….. Quadratic Formula: Discriminant b b 2 4ac x when given ax2 + bx + c = 0 2a a, b, c are the coefficients Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have. Determine the type and number of solutions: 1: x2 + 6x + 8 = 0 Practice: 1) 2x2 + 4x - 4 = 0 2: x2 + 6x + 9 = 0 2) 8x2 = 9x – 11 3: x2 + 6x = -10 3) 3x – 5x2 + 1 = 6 – 7x Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: N.CN.A.1 Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. N.CN.A.2 Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers Complex numbers Can you think of a number to make this equation true? Imaginary number: i= 1 i2 = -1 Examples: -25 -13 Examples: 2) –x2 + 4x = 5 1) 2x2 + 11 = -37 Complex Number Standard Form: a + bi (real part + imaginary part) Operations with complex numbers (+, -, x): Combine real parts and combine imaginary parts. 1) (8 - i) + (5 + 4i) 3) (7 - 6i) - (3 - 6i) 5) 6i(4 – 2i) 2) (6 + 7i) + (1 - 3i) 4) (3 + 7i) - (8 - 2i) Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variableMath- Aids.Com | Dynam ically Created Math Worksheets Part 6: Parabolas – Graphs of Quadratic Equations y = ax2 + bx + c these quadratic equations form a Safari Power Saver Click to Start Flash Plug-in shaped graph called a PARABOLA. Ex A) y = x2 Step 1: Identify a, b, c a= b= c= Step 2: Monday September 22, 2014 If “a” is positive, then parabola opens up . Vertex is a minimum If “a” is negative, then parabola opens down . Vertex is a maximum Worksheet Viewe You May Print or Save This Dynam To Print the Worksheet: Right Click on the If the Answer Key is Selected, Sc Step 3: Identify vertex (Xv, Yv) - ordered pair of turning point Xv (x coordinate) = b 2a To find Yv (y coordinate), substitute Xv into the equation. V= Sketch graph X y State x-intercepts: More instructions at the botto Safari Power Saver Click to Start Flash Plug-in 1: y = x2 + x – 12 a) a = b= c= b) which way will the graph open? Min or max? c) vertex: Monday September 22, 2014 d) sketch graph: X Worksheet Viewer P You May Print or Save This Dynam To Print the Worksheet: Right Click on the W If the Answer Key is Selected, Scr y More instructions at the bottom State x-intercepts: 2: y = 2x2 a) a= b= c= b) which way will the graph open? Min or max? c) vertex: http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin d) sketch graph X y State x-intercepts: http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin Math- Aids.Com | Dynam ically Created Math Worksheets 3: -x2 - 9x + 8 = 0 Safari Power Saver Click to Start Flash Plug-in a) a= b= c= b) which way will the graph open? Min or max? c) vertex: d) sketch graph Monday September 22, 2014 X y Worksheet Viewer P State x-intercepts: You May Print or Save This Dynam To Print the Worksheet: Right Click on the W If the Answer Key is Selected, Scr More instructions at the bottom 4: 2x2 – 8x + 6 = 0 a) a= b= c= b) which way will the graph open? Min or max? http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin c) vertex: d) sketch graph X y State x-intercepts: http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin Math- Aids.Com | Dynam ically Created Math Worksheets Safari Power Saver Click to Start Flash Plug-in 5: x2 – 2x - 1 = 0 a) a= b= c= b) which way will the graph open? Min or max? c) vertex: d) sketch graph Monday September 22, 2014 Worksheet Viewer P X y State x-intercepts: You May Print or Save This Dynam To Print the Worksheet: Right Click on the W If the Answer Key is Selected, Scr More instructions at the bottom 6: 3x2 – 18x + 20 = 0 a) a= b= c= b) which way will the graph open? Min or max? http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin c) vertex: d) sketch graph X y State x-intercepts: http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable Word problems: 1) A ball is thrown, its height after t seconds is given by the function h(t) = 7 + 6t – t2 , where t is the time in seconds after the object is propelled and h is the object’s height in feet. a) Draw a picture of the situation. b) How many seconds after the ball is thrown will the object hit the ground? c) What is the time for it to reach maximum height? d) What is the maximum height? e) What is the height at 2 seconds? 2) Given the area of a patio, sketch and label a picture with a possible expression for the width and length. a) Area = 3x2 – 15x b) Area = 9 + 54x2 Is it possible to build the patio? Yes No Explain: Is it possible to build the patio? Yes No Explain: Algebra 2 Solve Quadratic Equations using Factoring - Notes Monday September 22, 2014 Worksheet Viewe Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ You May Print or Save This Dynam To Print the Worksheet: Right Click on the NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable If the Answer Key is Selected, Sc Part 6: Parabolas – Graphs of Quadratic Equations Graph using the calculator. More instructions at the botto Ex A) y = x2 Step 1: load the equation into “y =” (remember to clear out all equations in there) Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Step 3: fill in some values for table and state x- intercepts x y Step 4: sketch graph. http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor 1: y = x2 + 1 Step 1: load the equation Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point Math- Aids.Com | Dynam ically Created Math Worksheets V( , ) is the vertex a min or max? What is the value of the min or max? Safari Power Saver Click to Start Flash Plug-in Step 3: fill in some values for table and state x- intercepts x y Monday September 22, 2014 Worksheet Viewer P You May Print or Save This Dynam To Print the Worksheet: Right Click on the W If the Answer Key is Selected, Scr More instructions at the bottom Step 4: sketch graph. 2: y = -x2 + 2 Step 1: load the equation http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Step 3: fill in some values for table and state x- intercepts x y Step 4: sketch graph. 3: y = x2 - 2x Step 1: load the equation Math- Aids.Com | Dynam ically Created Math Worksheets Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Safari Power Saver Click to Start Flash Plug-in Step 3: fill in some values for table and state x- intercepts x y Monday September 22, 2014 Worksheet Viewe You May Print or Save This Dynam To Print the Worksheet: Right Click on the If the Answer Key is Selected, Sc More instructions at the botto Step 4: sketch graph. 4: y = -2x2 – 4x + 7 http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor Step 1: load the equation Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Step 3: fill in some values for table and state x- intercepts x y Step 4: sketch graph. 5: y = -x2 + 3 Step 1: load the equation Math- Aids.Com | Dynam ically Created Math Worksheets Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Safari Power Saver Click to Start Flash Plug-in Step 3: fill in some values for table and state x- intercepts x y Monday September 22, 2014 Worksheet Viewe You May Print or Save This Dynam To Print the Worksheet: Right Click on the If the Answer Key is Selected, Sc More instructions at the botto Step 4: sketch graph. 6: y = .33x2 – 2x + 4 Step 1: load the equation http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point V( , ) is the vertex a min or max? What is the value of the min or max? Step 3: fill in some values for table and state x- intercepts x y Step 4: sketch graph. Algebra 2 Solve Quadratic Equations using Factoring - Notes Name:_______________________________________ Date:_____________ _______________________________________________________________________________________ NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable Completing the Square When solving a Quadratic Equation you need to factor it first. This is a short cut to factoring. If you have a Perfect Square Trinomial (PST) then: x2 + 2x + 1 = (x + 1)2 = (x + 1)(x + 1) How do you create a PST? Take half the 2nd term and then square it. 1) x2 + 12x + _____ 2) x2 + 14x + _____ 3) x2 - 20x + _____ Solve the Quadratic Equation. 1) x2 + 12x + 36 = 0 2) x2 + 14x + 49 = 0 3) x2 - 20x + 100 = 0 Solve the Quadratic Equation by completing the square. Goal: Try to make a PST. Step 1: Is left side x2 + bx? NO - move all numbers to right. Step 2: Add the number to make a PST to BOTH sides. Step 3: Factor Step 4: Take square root of BOTH sides (one = +, one = -) Step 5: solve 1) x2 + 2x = 0 2) x2 + 6x = 16 3) x2 - 2x – 8 = 0 4) x2 + 8x + 10 = 0