H28 VIERKANTSVERGELIJKINGEN

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H8 VIERKANTSVERGELIJKINGEN vwo 8.0 INTRO - - 8.

1 H8 VIERKANTSVERGELIJKINGEN vwo 8.0 INTRO TERUGBLIKKEN 3 a x = 3½ b x + 7 = x + 7 = x + 6 = x Dus x = 3 c x = of x = - d x + 6 = of x + 6 = - x= - of x = -0 e Er is geen olossing, want het kwadraat van een getal kan niet negatief zijn. f x = 7 of x = - 7 g x + = 7 of x + = - 7 x = of x = - 7 h x x = 0 x(x ) = 0 x = 0 of x = i x + x = 0 x(x + ) = 0 x = 0 of x = - j x x = 0 x(x ) = 0 x = 0 of x = k x = 7 l x + = 7, dus x = 6 m =, dus x = 7, dus x = 3½ x 7 n x = 7 o = 3½, dus x = = x 3 7 =, dus x = 7, dus x = 3½ x 7 q x = 9 r Er is geen olossing, want de wortel van een getal kan niet negatief zijn. s x + = 9, dus x = 8 t x = 7 x = = 7 of x = = a x + x + 36 = 6 x + x + 0 = 0 b x + x + 0 = (x + )(x + 0) c x = - of x = -0 a - = - - = - - = - - = = = - -6 = = - b x + x = (x + 8)(x 3) x + x = 0 x = -8 of x = 3 6 a (x + )(x ) = 0, dus x = - of x = b (x + 6)(x ) = 0, dus x = -6 of x = c (x + 6)(x + ) = 0, dus x = -6 of x = - d x(x + ) = 0, dus x = 0 of x = - 7 a x + x 8 = 0 (x + 8)(x 6) = 0 x = -8 of x = 6 b x x = 0 x(x ) = 0 x = 0 of x = c (x + x 3) = 0 (x + 3)(x ) = 0 x = -3 of x = d -x + = -60 x = 6 x = 8 of x = -8 e x = x + 0 x x 0 = 0 (x 0)(x + ) = 0 x = 0 of x = - f x x + 3x 3 = 7 x + x 0 = 0 (x + )(x 0) = 0 x = - of x = 0 g x(x + x 3) = 0 x(x + 3)(x ) = 0 x = 0 of x = -3 of x = h x x = 0 x (x ) = 0 x = 0 of x = i x + x + = x + 3 x + x = 0 (x + )(x ) = 0 x = - of x = j x (x + x + ) = 0 x (x + ) = 0 x = 0 of x = - 8. KWADRAATAFSPLITSEN 8 a x + 3x + 3x = x + 6x b x + 6x = 7 x + 6x 7 = 0 (x + 7)(x ) = 0 x = -7 of x = Dus x =, want x kan niet negatief zijn. c x + 6x = 6 x + 6x 6 = 0 (x + 8)(x ) = 0 x = -8 of x = Dus x =, want x kan niet negatief zijn. H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode

2 d ( ) + 6( ) = = 0 Klot. e 9 f zijde is x + 3; oervlakte = (x + 3) g x + 6x = (x + 3) 9 h x + 6x = (x + 3) 9 = (x + 3) = 0 9 a x + b x + 3 = 0 of x + 3 = - 0 x = of x = -3 0 c x = - + of x = - d (x + ) = 37 e x = of x = a x + x = (x + 6) 36 b x + x = (x + 6) 36 = (x + 6) = 0 x + 6 = 0 of x + 6 = - 0 x = of x = -6 0 c x + x + = 0 (x + 6) 36 + = 0 (x + 6) = 3 x + 6 = 3 of x + 6 = - 3 x = -6 + of x = -6 a x 0x = (x 0) 00 b x 7x = (x 3½) ¼ c x 8x = (x ) 6 d x + x = (x + ½) 30¼ e x x = (x 0½) 0¼ f x + x = (x + ½) ¼ g x x = (x ½) ¼ 8.3 VIERKANTSVERGELIJKINGEN OPLOSSEN a -x + 3x = x x 3x = -x + x + x = (x + ) = (x + ) = = x = - + of x = - (NB: = ) b x = x + 6 x = x + 3 x x = 3 (x ) = 3 (x ) = x = of x = - x = 3 of x = - Handiger: x x 3 = 0 (x 3)(x + ) = 0 x = 3 of x = - c (x + ) = -(x + ) + 7 x + x + = -x + 7 x + 3x = 0 (x + )(x ) = 0 x = - of x = d x + x + 3 = 0 (x + ) = 0 (x + ) 3 = x + = 3 of x + = 3 x = of x = - 3 e 3x + 6x + 9 = 0 x + x + 3 = 0 (x + ) + 3 = 0 (x + ) = - Er zijn geen olossingen. f -x x = 0 -x x 0 = 0 x + x + 0 = 0 (x + ) + 0 = 0 (x + ) = -9 Er zijn geen olossingen. g (x) = x x x + = 0 x x + ¼ = 0 (x ½) = 0 x = ½ 8. KRUISLINGS VERMENIGVULDIGEN 3 a x + b 3 a x + = 3x x = b Beide kanten met x + vermenigvuldigen geeft: x + 3 = (x +) - = x x = -½ c x + = x x = x = of x = - b x = x + 3x x + x + = 0 (x + ) = 0 x = - H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode

a (x ) = (x + ) x = x + x = 6 x = 3 b x(x ) = (x + )(x + ) x x = x + 3x + x + x + = 0 (x + ) = 3 x + = 3 of x + = - 3 x = - + 3 of x = - 3 c (x + ) = x x + = x of x + = -x x = of 3x = - x = of x = -⅓

3 a (x ) = (x + ) x = x + x = 6 x = 3 b x(x ) = (x + )(x + ) x x = x + 3x + x + x + = 0 (x + ) = 3 x + = 3 of x + = - 3 x = of x = - 3 c (x + ) = x x + = x of x + = -x x = of 3x = - x = of x = -⅓ d 3( x) = x 3 3x = x x = x =, maar let o, zie het volgende!! e De linkerkant wordt dan bijvoorbeeld 0 3 en dit heeft geen betekenis. f x = -½ en x = ; x = 0 en x = - g De waarden 0 en. h x = x x x x = 0 x(x ) = 0 x = 0 of x = Maar x = 0 maakt noemers 0, dus de enige olossing is x =. 6 a groot = x klein = hele lijnstuk = x + Dus groot : klein = hele lijnstuk : groot wordt x x+ dan x : = x + : x. Dus =. x b x = x + x x = 0 (x ) = 0 (x ) = x = = of x = - x = + of x = Dus het gulden getal is + = 8. CIRKELS 7 9,6 6,7 =,9 hm, 9,6 =,8 hm 8 a ; ; 3 b -7 en c = - +. Als x <, dan is de afstand van x tot : x. f als x > 0, dan is de afstand van x tot 0: x. als x < 0, dan is de afstand van x tot 0: -x. 9 a 7 + = 0 = 0 a a b OB = + = OC = + 7 = OD = + = OE = + = c Een cirkel met middelunt O(0,0) en straal. b (-x) = x en (-y) = y P b (-) + = 0, klot. r = 0 = c Zie onderdeel a. d Dan y = 0, dus x = 0, dus x = of x = -. Dus (,0) en (-,0). d 7 ; 7 + ; 7 e Als x >, dan is de afstand van x tot : x. H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode 3

a a b x + y = 8 en x + y = 3 a r = b a b AC = 6; BC = c 88 d a b; b a c (3,), (-3,), (-,3), (,0), (0,-), enzovoort.

c x = of x = - Snijunten (, ) en (-,- ). f Zie onderdeel b.

i Snijunt is (a,a + ) a + (a + ) = a + a + 0a + = a + 0a = 0 a(a + ) = 0 a = 0 of a = - Snijunten (0,) en (-,-3).

4 a a b x + y = 8 en x + y = 3 a r = b a b AC = 6; BC = c 88 d a b; b a c (3,), (-3,), (-,3), (,0), (0,-), enzovoort. d Zie onderdeel b. e x + x =, oftewel x = x = b Omdat tegengestelde getallen hetzelfde kwadraat hebben. c x = of x = - Snijunten (, ) en (-,- ). f Zie onderdeel b. g a + a + a + = a + a = 0 a + a = 0 (a + )(a 3) = 0 a = - of a = 3 Snijunten (-,-3) en (3,). h Zie onderdeel b. i Snijunt is (a,a + ) a + (a + ) = a + a + 0a + = a + 0a = 0 a(a + ) = 0 a = 0 of a = - Snijunten (0,) en (-,-3). j (0,0) k niet één d (3 ) + (y + ) = 3 (y + ) = y = -+ of y = - Snijunten (3,- + 3 ) en (3,- 3 ). e y = 0, dus (x ) + = 3 (x ) = x = + 3 of x = 3 Snijunten ( + 3,0) en ( 3,0). H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode

6 a b - c - d x + 3; y e (x + 3) + ( y) = 9 7 C : (x 3) + (y 3) = 9 C : (x + ) + (y ) = C 3 : (x + 3) + (y + ) = 3 C : (x ) + (y + 3) = 8 8 30 a (x + ) ; (y 6) 36 b (x + ) + (y 6) 36 = 39 (x + ) + (y

5 6 a b - c - d x + 3; y e (x + 3) + ( y) = 9 7 C : (x 3) + (y 3) = 9 C : (x + ) + (y ) = C 3 : (x + 3) + (y + ) = 3 C : (x ) + (y + 3) = a (x + ) ; (y 6) 36 b (x + ) + (y 6) 36 = 39 (x + ) + (y 6) = = 00 c Middelunt (-,6) en straal 0. 3 x + x = (x + ) en y y = (y ) 6 Dus: x +y +x y + 8 = 0 (x + ) + (y ) (x + ) + (y ) = + Middelunt (-, = = ) en straal is 8.6 GEMENGDE OPGAVEN 3 a (x 3) 9 + (y 3) 9 + = 0 (x 3) + (y 3) = 7 Middelunt (3,3) en straal 7. b =. 9 a c Zie onderdeel b. d (x 3) + (3x + 3) = 7 (x 3) + (3x + ) = 7 0x + 6x + 3 = 7 x + 3 x = 0 (x ) 9 = 00 x = 0 of x = - Snijunten (, 6 ) en (-,). b (x ) + (x + ) = 0 x 8x x = 0 x 8x + 6 = 0 x x + 3 = 0 (x )(x 3) = 0 x = of x = 3 Als x =, dan y = + = 3 Als x = 3, dan y = 3 + = Snijunten: (,3) en (3,). c x 8x y y + = 0 x + y 8x y + 0 = 0 33 a (60 x) = x = 0 of 60 x = -0 x = 30 of x = 30 + Maar x < 30, dus x = 30 b (60 x) = x (60 x) x + x = -8x + 0x x 80x = 0 x 0x = 0 (x 0)(x 30) = 0 x = 0 of x = 30, dus x = 0 H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode

6 3 a x + 6 = x + x x x 6 = 0 (x 3)(x + ) = 0 x = 3 of x = - Geen van beide olossingen maken noemers 0, dus de olossingen zijn x = 3 en x = -. b x + = x + x x x = 0 (x )(x + ) = 0 x = of x = - x = - maakt noemers 0, dus de enige olossing is. 3 a C = (6 ½) ½ 0 = b C = (6 x) x 0 = 60x 0x c 60x 0x = 0 -x + 6x = 0 x 3x + = 0 (x ½) ¼ + = 0 (x ½) = ¼ = x = + of x = x 3 36 a In de kleine: = tanα x 8 In de grote: = tanα x + 7 x 3 8 b = x x + 7 (x 3)(x + 7) = 8x x + x = 8x x x = 0 (x 7)(x + 3) = 0 x = 7 of x = -3, dus x = 7 37 a 36 (6 x) x = 36 (36 x + x ) x = x x x = x x b x x = x 6x + = 0 (x 3) 9 + = 0 (x 3) = 8 x 3 = of x 3 = - x = 3 + of x = 3 Maar x < 3, dus x = a Border: x + 3 x = x + x Dus: x + x = 6 x + 6x 8 = 0 (x + 3) = 7 x = of x = -3 7 Maar x > 0, dus x = b (x + x) = 6 x + 6x = (x + 3) = 3 x = of x = -3 3 Maar x > 0, dus x = Oker 6 a - b x + x + 8x + 6 x + 8x + (x + 3)(x + ) y + 0 y + 0y + 00 y + 0y + 99 (y + 9)(y + ) z + z + y + z + z + 0 (z + 0)(z + ) ( + ) c - d (x + a) = (x + a ) (x + a + ) 7 a x x = 0 (x + 3)(x 7) = 0 x = -3 of x = 7 b x x = 9 x x = 0 Dus ook nu geldt x = -3 of x = 7 c x 3 x x = 0 x(x x ) = 0 x(x + 3)(x 7) = 0 x = 0 of x = -3 of x = 7 Als het getal een olossing is van ax + bx + c = 0, dan is het getal een olossing van cx + bx + a = 0. Voorbeeld Het getal 3 is een olossing van de vergelijking x x + 3 = 0. Door invullen kun je nagaan dat het getal 3 een olossing is van 3x x + = 0. Bewijs Stel het getal is een olossing van ax + bx + c = 0. Dus a + b + c = 0. We vullen het getal in de uitdrukking cx + bx + a in. We krijgen: c ( ) + b + a = ( ) (c + b + a ) Omdat a + b + c = 0 geldt: 0 = 0. c ( ) + b + a = ( ) Dus is een olossing van de vergelijking cx + bx + a = 0. H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode 6

EXTRA OPGAVEN a x x = x x = 0 (x 7)(x + ) = 0 x = 7 of x = - Beide olossingen voldoen.

7 3 a b Zie onderdeel a. c Zie onderdeel a. d WA = 90 en WB = 0 e Geldt WA + WB = AB? Ja, dus hoek W is recht. f Zie onderdeel a. g ( x ) + y h PA + PB = AB, dus (x + ) + y + (x ) + y = 00 i x + 0x + + y + x 0x + + y = 00 x + y = Middelunt (0,0) en straal. EXTRA OPGAVEN a x x = x x = 0 (x 7)(x + ) = 0 x = 7 of x = - Beide olossingen voldoen. b x x 7 = 0 x x 3½ = 0 (x ) ½ = 0 x = + of x = c x x = 0 (x ½) 6¼ = 0 (x ½) = ¼ x = + of x = d (x + ) = x + = ½ of x + = -½ x = ½ of x = -3½ Beide olossingen voldoen. e x + 8x + 0 = x x 8x 0 = 0 x x = 0 (x )(x + ) = 0 x = of x = - f x x + = 0 (x ) = -3 Er zijn geen olossingen. a 3 a g x = x + x x = 0 (x )(x + ) = 0 x = of x = - h (x + ½) = 36 x = ½ of x = -6½ i x + = 9 x = 8 j x = x + 3x x 3x = 0 x(x 3) = 0 x = 0 of x = 3 x = 0 maakt de noemer 0, dus de enige olossing is 3. b Zie onderdeel a. c rc k is 3, dus y = 3x + 8 cirkel: (x + ) + (y ) = Snijunten bealen: (x + ) + (3x + 8 ) = 0x + 6x + 3 = x +,6x +,8 = 0 (x +,3) =,9,8 = 0,9 Dus x = 0,7,3 = -,6 of x = -0,7,3 = -3 Snijunten (-,6 ; 3,) en (-3 ; -). b C: (x ) + (y ) = 9 Snijunt bealen: (x ) = x = + of x = A(,3) en B( +,3) en AB = 6x + (9⅓ x)x = 0 6x + 9⅓x x = 0 x ⅓x + 0 = 0 (x 3 3 ) = 69 9 x = = of x = = 3⅓ Maar x < 9⅓, dus x = 3⅓. H8 VIERKANTSVERGELIJKINGEN vwo de Wageningse Methode 7

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4 a x x + 36 = 16 x x + 20 = 0 b x x + 20 = (x + 2)(x + 10) c x = -2 of x = -10

4 a x x + 36 = 16 x x + 20 = 0 b x x + 20 = (x + 2)(x + 10) c x = -2 of x = -10 H8 VIERKANTSVERGELIJKINGEN VWO 8.0 INTRO - - 8. TERUGBLIKKEN a x = b x + 7 = x + 7 = x + 6 = x x = c x = of x = - d x + 6 = of x + 6 = - x = - of x = -0 e Er is geen olossing, want het kwadraat van een