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QUADRATICS JOURNAL

QUADRATICS JOURNAL. Amani Mubarak 9-5. How to factor polynomials. 1.First multiply aXc. 2.Now find two factors of that multiply to the answer of aXc, and that will also Add up to b.

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QUADRATICS JOURNAL

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  1. QUADRATICS JOURNAL Amani Mubarak 9-5

  2. How to factor polynomials 1.First multiply aXc. 2.Now find two factors of that multiply to the answer of aXc, and that will also Add up to b. *USE A T-TABLE TO MAKE IT EASIER. (In one side of t-table you write b and in the other side the answer of aXc.) Ex.1 2X²-8X+6 12 8 1. 2X6=12 -2 X -6 2 2 -2 + -6 X= -1, -3

  3. Ex.2 Ex.4 x² + 8x = -15 +15 +15 X²+8x+15= 0 15 8 3X5 3+5 x x= 3,5 3x²-2x-16= 0 -48 -2 -8X6 -8+6 3 3 X= 8/3, -2 Ex.3 x² +7x+15= 5 10 7 -5 2x5 2+5 x² + 7x + 15= 0 x x= 2,5

  4. QUADRATIC FUNCTION • A QuadraticFunctioniswritten in theform: f(x) = ax2 + bx + c. • The graph of a quadratic function is a curve called a parabola. LINEAR FUNCTION • A Linear Functioniswritten in theform:y= mx+ b • The graph of a linear function is a straight line.

  5. Itsvery simple totellthedifferencebetweenthistwotypes of functions, since a graph of a quadraticfunctionwillhave a curved line, parabola, and a linear functionisjust a straight line. Ex.1 Ex.2 y= 1/2x + 2 y=3x² + 6x + 1

  6. Ex. 4 Ex. 3 y= -x + 5 y= 1/2x² +0 + 0

  7. HOW TO GRAPH A QUADRATIC FUNCTION • f (x) = ax2 + bx + c • Y intercept of the graph is found by f(0)=c • X intercept of the graph is found by solving the equation: ax2 + bx + c = 0 • ax2 + bx + c = 0 is solved by using –b/2a • STEPS: 1. set = 0 2. graphthefunction a. make a t-table b. findthevertex x= -b÷2ª c. pick 2 pointstotheleft and 2 totheright. d. graphthe parable 3. find x-valueswhereitcrossesthe x-axis.

  8. Examples: • y= -x² + 0 + 0 -b/2ª= 0 x y 0 0 1 -1 2 -4 3 -9 X=0,0

  9. 2. y= 1/2x² +0 + 0 -b/2(a)=0 x y 0 0 1 0.5 2 2 3 4.5 X=0,0

  10. 3. y=3x² + 6x + 1 -b/2(a)= -6/2(3) -6/6= -1 X Y -1 -2 0 1 1 10 2 19 X= -1,-2

  11. 4. y= x² + 2x + 5 -b/2(a)= -2/2(0)= -2/2 x y -2 -3 -1 -2 0 -1 1 0

  12. Howtosolve a quadraticequation by graphingit • a (x + b)² + c= 0 a- changesthestepnessoftheline. b- moves rightotleft. Left= + Positive= - c- moves thevertexupordown. (Positive goesup. Negativegoesdown.) * If a is lessthan 0 if a isbiggerthan 0

  13. Examples: • Y= -2 (x-4)²+5 Y= 2(x+3) 2-2

  14. Y=4/2 (x-2) -6 Y=2/4(x+3) -3

  15. Howtosolvequadraticequationusingsquareroots • X²=k • Ifyourequation has a # nextto x: • 1. Youhaveto divide bothsides by that # toisolate x². • Thenyousimplify. • Use thesquarerootpropertytoobtainto posible answers.

  16. Examples: • K²=16 √ 16 k= 4, -4 2. K²=21 √ 21 k= 7,-7 3. 4n²= 20 4 N²=5,-5 7x² = -21 7 x²= 3, -3

  17. Howtosolvequadraticequationusingfactoring: • In ordertofactor a quadraticyoumustfindcommonnumbersthatwillmultiply b and add up to c. Thenputeach set = 0. Ex.1 Ex.2 x2 + 5x + 6 = (x + 2)(x + 3) (x + 2)(x + 3) = 0 x+ 2 = 0  or  x + 3 = 0 x = –2  or  x = – 3 x= –3, –2 x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 x – 3 = 0  or  x + 1 = 0 x = 3  or  x = –1 x= –1, 3

  18. Ex.3 x2 + 5x – 6 = 0 (x + 6)(x – 1) = 0 x + 6 = 0  or   x – 1 = 0 x = –6  or   x = 1 x= –6, 1 Ex.4 . x2+5x+6=0. (x+2)(x+3)=0. x=-2 and x=-3.

  19. Completingthesquare • To complete thesquare: • Get a=1 • Find b, divide b/2, squareit (b/2)² • Factor (x+b/2)² Ex. X² + 14x + 49 x²+26x+169

  20. Howtosolvequadraticequationsusingcompletingthesquare: • STEPS: 1. get x²=1 2. get c by itself 3. complethesquare 4. add b/2² tobothsides 5. squarerootbothsides

  21. Examples: 3.X² + 16p – 22= 0 +22 +22 • X² + 16x= 22 +1 • x+1= ± 4.8 • X= 5.8, 3.8 • 4. X² + 8k + 12 = 0 • -12 -12 • X² + 8x = 13 • X+1= ±3.6 • X= 4.6, 2.6 • A² + 2 a – 3= 0 +3 +3 A²+ 2 a= 3 +1 √(a+1)² = √4 A + 1= ± 2 A= 3,1 2. A² - 2a – 8= 0 +8 +8 A²- 2a= 8 +1 √(a-1)² = √9 A-1= ± 3 A= 4,2 √(x + 1)² = √23 √(x+1)² = √13

  22. Howtosolvequadraticequationsusingquadratic formula: • X= -b ± √b²-4ac 2 a 1. Find a, b, c and fillthem in. Ex.2 Ex.1 3x² -4x -9= 0 A= 3 b= -4= c=9 4± √16+108= 4±√124 6 6 2±√31 3 M²- 5m-14=0 A= 1 b=-5 c= -14 5 ± √ 25 + 56 = 5±√81 2 2 5±√9 2

  23. Ex.3 C²- 4c + 4= 0 4± √16-16 = 4± √0 -8 -8 2±√0 -4 Ex.4 3±√9+40 = 3±√49 4 4 3±√7 4

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