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 getal kan niet negatief zijn. f x = 7 of x = - 7 g x + = 7 of x + = - 7 x = - + 7 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 x = 6 m x 7 x = 7 x = n x = 7 o = x x = = 7 x 7 x = 7 x = q x = 9 r Er is geen olossing, want de wortel van een getal kan niet negatief zijn. s x + = 9 x = 8 t x = 7 x of x - - 7 7 7 7 7 7 a x + x + 6 = 6 x + x + 0 = 0 b x + x + 0 = (x + )(x + 0) c x = - of x = -0 a - = - - = - - = - - = - -8 = - - 8 = - -6 = - - 6 = - b x + x = (x + 8)(x ) x + x = 0 (x + 8)(x ) = 0 x = -8 of x = 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 ) = 0 (x + )(x ) = 0 x = - 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 + x = 7 x + x 0 = 0 (x + )(x 0) = 0 x = - of x = 0 g x(x + x ) = 0 x(x + )(x ) = 0 x = 0 of x = - of x = h x x = 0 x (x ) = 0 x = 0 of x = i x + x + = x + 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 + x + x = 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. de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO
c x + 6x = 6 x + 6x 6 = 0 (x + 8)(x ) = 0 x = -8 of x = Dus x =, want x kan niet negatief zijn. d (- + 9 ) + 6(- + 9 ) = 9 6 9 + 9 8 + 6 9 = 0, klot. e 9 f zijde = x + ; oervlakte = (x + ) g x + 6x = (x + ) 9 h x + 6x = (x + ) 9 = (x + ) = 0 x + = 0 of x + = - 0 x = - + of x = - 9 a x + b c x = - + of x = - d (x + ) = 7 e x = - + 7 of x = - 7 0 a x + x = (x + 6) 6 b x + x = (x + 6) 6 = (x + 6) = 0 x + 6 = 0 of x + 6 = - 0 x = -6 + 0 of x = -6 0 c x + x + = 0 (x + 6) 6 + = 0 (x + 6) = x + 6 = of x + 6 = - x = -6 + of x = -6 a x 0x = (x 0) 00 b x 7x = (x ) c x 8x = (x ) 6 d x + x = (x + ) 0 e x x = (x 0 ) 0 f x + x = (x + ) g x x = (x ) 8. VIERKANTSVERGELIJKINGEN OPLOSSEN a -x + x = x x x = -x + x + x = (x + ) = (x + ) = = x = - + of x = - (NB: ) b x = x + 6 x = x + x x = (x ) = (x ) = x = of x = - x = of x = - Handiger: x x = 0 (x )(x + ) = 0 x = of x = - c (x + ) = -(x + ) + 7 x + x + = -x + 7 x + x = 0 (x + )(x ) = 0 x = - of x = d x + x + = 0 (x + ) 6 + = 0 (x + ) = x + = of x + = - x = - + of x = - e x + 6x + 9 = 0 x + x + = 0 (x + ) + = 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 + = 0 (x ) = 0 x = 8. KRUISLINGS VERMENIGVULDIGEN a x + b a x + = x x = b Beide kanten met x + vermenigvuldigen geeft: x + = (x + ) x + = x + - = x x = - c x + = x x = x = of x = - de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO
d x = x + x x + x + = 0 (x + ) = 0 x = - a (x ) = (x + ) x = x + x = 6 x = b x(x ) = (x + )(x + ) x x = x + x + x + x + = 0 (x + ) = x + = of x + = - x = - + of x = - c (x + ) = x x + = x of x + = -x x = of x = - x = of x = - d ( x) = x x = x x = x =, maar let o, zie het antwoord bij e!! e De linkerkant wordt dan bijvoorbeeld 0 en dit heeft geen betekenis. f x = - 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 dan x : = (x + ) : x. Dus x x. x b x = x + x x = 0 (x ) = 0 (x ) = 8 a ; ; b -7 en c d 7 ; 7 + ; 7 e Als x >, dan is de afstand van x tot : x. 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 b OB = OC = OD = OE = 0 a AO 7 0 7 c Een cirkel met middelunt O(0,0) en straal. ac b (-x) = x en (-y) = y x = of x = - x = + of x = Dus het gulden getal is + =. 8. CIRKELS 7 9,6 6,7 =,9 hm, 9,6 =,8 hm b (-) + = 0, klot. r = 0 d Dan y = 0, dus x = 0, dus x = 0 of x = - 0 -. Dus (,0) en (-,0). de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO
a a b AC = 6 ; BC = c 87 + 0 = 88 d a b ; b a b x + y = 8 en x + y = a a r bdfh c (,), (-,), (-,), (,0), (0,-), enzovoort. e x + x =, oftewel x = x = 0 x = 0 0 of x = - Snijunten (, ) en (-,- ). g a + a + a + = a + a = 0 a + a = 0 (a + )(a ) = 0 a = - of a = Als a = -, dan y = a + = - + = -. Als a =, dan y = a + = + =. Snijunten (-,-) en (,). i Snijunt is (a,a + ) a + (a + ) = a + a + 0a + = a + 0a = 0 a(a + ) = 0 a = 0 of a = - Snijunten (0,) en (-,-). j (0,0) k Niet één. b Omdat tegengestelde getallen hetzelfde kwadraat hebben. c d ( ) + (y + ) = (y + ) = y + = of y + = - - y = - of y = - Snijunten (,- ) en (,- ). e y = 0, dus (x ) + = (x ) = x = of x = - x = + of x = Snijunten (+,0) en (,0). de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO
6 a 7 8 b - c - d x + ; y e (x + ) + ( y) = 9 en dat is gelijk aan (x + ) + (y ) = 9 M C (,) en r = = 9 (x ) + (y ) = 9 M C (-,) en r = + = (x + ) + (y ) = M C (-,-) en r = + = (x + ) + (y + ) = M C (,-) en r = + = 8 (x ) + (y + ) = 8 b (x ) + (x + ) = 0 x 8x + 6 + x = 0 x 8x + 6 = 0 x x + = 0 (x )(x ) = 0 x = of x = Als x =, dan y = + =. Als x =, dan y = + =. Snijunten: (,) en (,). c x 8x + 6 + y y + = 0 x + y 8x y + 0 = 0 0 a (x + ) ; (y 6) 6 b (x + ) + (y 6) 6 = 9 (x + ) + (y 6) = 9 + + 6 = 00 c Middelunt (-,6) en straal 00 0. x + x = (x + ) en y y = (y ) 6 Dus: x + y + x y + 8 = 0 (x + ) + (y ) 6 + 8 = 0 (x + ) + (y ) = + 6 8 = Middelunt (-, ) en straal. 8.6 GEMENGDE OPGAVEN a (x ) 9 + (y ) 9 + = 0 (x ) + (y ) = 7 Middelunt (,) en straal 7. bc 9 a d (x ) + (x + ) = 7 (x ) + (x + ) = 7 x 6x + 9 + 9x + x + = 7 0x + 6x + = 7 x + x = 0 (x + 0 ) = 9 00 x + of x + = 9 7 0 00 0 x = of x = - 0 Als x =, dan y = + = 6. Als x = -, dan y = - + =. Snijunten (, 6 ) en (-,). - - = 9 7 0 00 0 de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO
a (60 x) = 00 60 x = 0 of 60 x = -0 x = 0 of x = 0 + Maar x < 0, dus alleen x = 0 voldoet. b (60 x) = x (60 x) 600 0x + x = 0x 8x x 80x + 600 = 0 x 0x + 00 = 0 (x 0)(x 0) = 0 x = 0 of x = 0 Maar x < 0, dus alleen x = 0 voldoet. a (x + ) = x(x + ) x + 6 = x + x x x 6 = 0 (x )(x + ) = 0 x = of x = - Geen van beide olossingen maken noemers 0, dus de olossingen zijn x = en x = -. b (x + ) = (x + x) x + = x + x x x = 0 (x )(x + ) = 0 x = of x = - x = - maakt noemers 0, dus de enige olossing is x =. a C = (6 ) 0 = b C = (6 x) x 0 = 60x 0x c 60x 0x = 0 -x + 6x = 0 x x + = 0 (x ) + = 0 (x ) = = x = of x = Allebei de olossingen voldoen. x 6 a In de kleine: tanα x 8 In de grote: tanα x 7 x 8 b x x 7 (x )(x + 7) = 8x x + x = 8x x x = 0 (x 7)(x + ) = 0 x = 7 of x = - Maar x > 0, dus alleen x = 7 voldoet. 7 a 6 (6 x) x = 6 (6 x + x ) x = 6 6 + x x x = x x b x x = 0 = x x + x 6x + = 0 (x ) 9 + = 0 (x ) = 8 x = 8 of x = - x = + of x = Maar x <, dus alleen x = voldoet. 8 a Border: x + x = x + x Dus: x + x = 6 x + 6x 8 = 0 (x + ) = 7 x = - + 7 of x = - 7 Maar x > 0, dus alleen x = - + 7 voldoet. b (x + x) = 6 x + x = 8 x + 6x = (x + ) 9 = (x + ) = x = - + of x = - Maar x > 0, dus alleen x = - + voldoet. SUPER OPGAVEN 6 a - b x + x + 8x + 6 x + 8x + (x + )(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 + )(x 7) = 0 x = - of x = 7 b x x = 9 x x = 0 Dus ook nu geldt x = - of x = 7. c x x x = 0 x(x x ) = 0 x(x + )(x 7) = 0 x = 0 of x = - of x = 7 de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO 6
Als het getal een olossing is van ax + bx + c = 0, dan is het getal een olossing van cx + bx + a = 0. abcf Voorbeeld Het getal is een olossing van de vergelijking x x + = 0. Door invullen kun je nagaan dat het getal een olossing is van x 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: c + b + a = 0 = 0. Dus is een olossing van de vergelijking cx + bx + a = 0. 8.8 EXTRA OPGAVEN a x(x ) = 7 x x = x x = 0 (x 7)(x + ) = 0 x = 7 of x = - Beide olossingen voldoen. b x x 7 = 0 x x = 0 (x ) = 0 (x ) = = 8 x = 8 8 of x = - x = + of x = c x x = 0 (x ) 6 = 0 (x ) = = x = of x = - x = of x = d (x + ) = (x + ) = x + = of x + = - d WA = 9 90 0 en WB = 0 e Geldt WA + WB = AB? Ja, want 90 0 90 0 00 0, dus hoek W is recht. g ( x ) y h PA + PB = AB, dus (x + ) + y + (x ) + y = 00 i x + 0x + + y + x 0x + + y = 00 x + y + 0 = 00 x + y = Middelunt (0,0) en straal. x = of x = - 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 ) = - Er zijn geen olossingen. g x = x + x x = 0 (x )(x + ) = 0 x = of x = -, maar x = - mag niet. h (x + ) = (x + ) = 6 x + = 6 of x + = -6 x = of x = -6 i x + = 9 x = 8 j x = (x + x) x x = 0 x(x ) = 0 x = 0 of x = x = 0 maakt de noemer 0, dus de enige olossing is x =. de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO 7
ab a c Vergelijking k: rck = - b = + = 8 dus y = x + 8 - - Vergelijking cirkel: (x + ) + (y ) = = Snijunten bealen: (x + ) + (x + 8 ) = 0x + 6x + = x +,6x +,8 = 0 (x +,) =,9,8 = 0,9 x = 0,9, = -,6 of x = - 0,9, = - Als x = -,6, dan y = -,6 + 8 =,. Als x = -, dan y = - + 8 = -. Snijunten (-,6 ;,) en (-,-). b C: (x ) + (y ) = 9 Snijunt bealen: (x ) = x = + of x = AB = ( ) 6x + (9 x)x = 0 6x + 9 x x = 0 x x + 0 = 0 (x ) = 69 9 x = + = of x = = Maar x < 6, dus x = is de enige olossing. de Wageningse Methode Antwoorden H8 VIERKANTSVERGELIJKINGEN VWO 8