( Wiskunde - Getal en Ruimte 12de editie, VWO 2 , H7 Kwadratische vergelijkingen)
Gemengde Opgaven - deel 1
(Opgave 1 t/m 4)
Opgave 1.
a. 5x² – 6x = 5∙x∙x – 6∙x
= x(5x – 6) ← Gemeenschappelijke factor x
b. 4x² – 24x + 20 = 4∙x² – 4∙6∙x + 4∙5 ← Gemeenschappelijke factor 4
= 4(x² –6x + 5)
–5
–1
= 4(x – 5)(x – 1) ← De Product-som-methode: –5×–1 =5 en –5 +(–1) = –6
c. 25x² – 49 = 5²x² – 7²
= (5x – 7)(5x + 7) ← De regel a² – b² = (a – b)(a + b)
d. 5ax – 25bx = 5∙a∙x – 5∙5∙b∙x
= 5x(a – 5b) ← Gemeenschappelijke factor 5 en x
e. 16x² + 4x = 4∙4∙x∙x + 4∙x
= 4x(4x + 1) ← Gemeenschappelijke factor 4 en x
f. 8x³y + 4x²y² – 12xy³ = 4∙2∙x∙x²y + 4∙x∙x∙y∙y – 3∙4∙x∙y∙y²
= 4xy(2x² + xy – 3y²) ← Gemeenschappelijke factor 4, x en y
g. x² – 3x + 2
–2 –1
= (x – 2)(x – 1) ← De Product-som-methode: –2 × –1 = 2 en –2 + (–1) = –3
h. 2x – x² = 2∙x – x∙x
= x(2 – x) ← Gemeenschappelijke factor x
i. x⁴ – x² = x²∙x² – x² ← Gemeenschappelijke factor x²
= x²(x² – 1²)
= x² (x – 1)(x +1) ← De regel a² – b² = (a – b)(a + b)
Opgave 2.
a. 49x² – 70x + 25 = 7²x² – 2(7x)(5) + 5²
= (7x – 5)² ← De regel (a – b)² = a² – 2ab + b²
b. 3x² + 54x + 243 = 3∙x² + 3∙18∙x + 3∙81
= 3(x² + 18x + 81) ← Gemeenschappelijke factor 3
= 3(x² + 2(9)(x) + 9²)
= 3(x + 9)² ← De regel (a + b)² = a² + 2ab + b²
c. x⁴ – 10x² + 9
–9
–1
= (x² – 9)(x² – 1) ← De Product-som-methode: –9 × –1 = 9 en –9 + (–1) = –10
= (x² – 3²)(x² – 1²)
= (x – 3)(x + 3)(x – 1)(x + 1) ← De regel a² – b² = (a – b)(a + b)
d. 4x⁴ – 24x² – 108 = 4∙x⁴ – 4∙6∙x² – 4∙27
= 4(x⁴ – 6x² – 27) ← Gemeenschappelijke factor 4
3
–9
= 4(x² + 3)(x² – 9) ← De Product-som-methode: 3 × –9 = –27 en 3 + (–9) = –6
= 4(x² + 3)(x² – 3²)
= 4(x² + 3)(x – 3)(x +3) ← De regel a² – b² = (a – b)(a + b)
e. 5x⁸ + 15x⁴ – 20 = 5∙x⁸ + 5∙3∙x⁴ – 5∙4
= 5(x⁸ + 3x⁴ – 4) ← Gemeenschappelijke factor 5
+4
–1
= 5(x⁴ + 4)(x⁴ – 1) ← De Product-som-methode: 4 × –1 = –4 en 4 + (–1) = 3
= 5(x⁴ + 4)((x²)² – 1²)
= 5(x⁴ + 4)(x² – 1)(x² + 1) ← De regel a² – b² = (a – b)(a + b)
= 5(x⁴ + 4)(x² – 1²)(x² + 1)
= 5(x⁴ + 4)(x – 1)(x + 1)(x² + 1) ← De regel a² – b² = (a – b)(a + b)
f. 3x⁵ – 39x³ + 108x = 3∙x∙x⁴ – 3∙13∙x∙x² + 3∙36∙x
= 3x(x⁴ – 13x² + 36) ← Gemeenschappelijke factor 3
– 9
– 4
= 3x(x² – 9)(x² – 4) ← De Product-som-methode: –9 × –4 = 36 en –9 + (–4) = –13
= 3x(x² – 3²)(x² – 2²)
= 3x(x – 3)(x + 3)(x – 2)(x + 2) ← De regel a² – b² = (a – b)(a + b)
Opgave 3.
a. x² – 81 = 0
x² – 9² =0
(x – 9)(x + 9) = 0 ← De regel a² – b² = (a – b)(a + b)
x – 9 = 0 of x + 9 = 0 → x = 9 ∨ x = – 9
b. x² + 81 = 0 ← Geen gemeenschappelijke factor
x² = – 81 → Geen oplossing, want een kwadraat is niet negatief.
c. x² +81x = 0
x∙x + 81∙x = 0
x(x + 81) = 0 ← Gemeenschappelijke factor
x =0 of x + 81 = 0 → x =0 ∨ x = – 81
d. x² + 81x + 80 = 0
+80
+1
(x + 80)(x + 1) = 0 ← De Product-som-methode: 80 × 1 = 80 en 80 + 1 = 81
x + 80 = 0 of x + 1 = 0 → x = –80 ∨ x = – 1
e. x² – 80x – 81 = 0
– 81
+1
(x – 81)(x + 1) = 0 ← De Product-som-methode: –81 × 1 = –81 en –81 + 1 = –80
x – 81 = 0 of x + 1 = 0 → x = 81 ∨ x = – 1
f. x² – 18x + 81 = 0
↓ ↓
– 2∙(x)∙(9) 9² ← De regel (a – b)² = a² – 2ab + b²
(x – 9)² = 0 → x = 9
g. (x +1)(x – 81) = 0 → x = – 1 ∨ x = 81
h. x(x +81) = 0 → x = 0 ∨ x = – 81
i. x² – 18x +18 = 81
– 81 = – 81
x² – 18x – 63 = 0 ← Maak het rechterlid nul.
–21
+3
(x – 21)(x +3) = 0 ← De Product-som-methode: –21 × 3 = –63 en –21 + 3 = –18
x – 21 = 0 of x +3 = 0 → x =21 ∨ x = –3
Opgave 4.
a. (2x – 1)(6x + 3) = 0
2x – 1 = 0 of 6x + 3 = 0 → x = ½ ∨ x = – ½
b. 3a² + 1 = 28
– 28 = –28
3a² – 27 = 0 ← Maak het rechterlid nul.
3∙a² – 3∙9 = 0 ← Gemeenschappelijke factor 3
3(a² – 9) = 0
3(a² – 3²) = 0
3(a – 3)(a + 3) = 0 ← De regel a²– b² = (a + b)(a – b)
a – 3 of a +3 = 0 → a =3 ∨ a = –3
c. 5a² – 20a = 0
5∙a∙a – 4∙5∙a = 0
5a(a – 4) = 0 ← Gemeenschappelijke factor 5 en a
a = 0 of a – 4 = 0 → a =0 ∨ a = 4
d. p² + 21 = 10p – 4
–10p +4 = –10p +4
p² –10p + 25 = 0 ← Maak het rechterlid nul.
p² – 2∙(p)∙(5) + 5² =0
(p – 5)² = 0 ← De regel (a – b)² = a² – 2ab + b²
p – 5 = 0 → p = 5
e. 144 – 49q² = 0
× (–1) = ×(–1)
–144 + 49q² = 0 ← Vermenigvuldig beide zijden met –1. ※ 0×(–1) = 0
49q² – 144 = 0
7²q² – 12² = 0
(7q – 12)(7q + 12) = 0 ← De regel a²– b² = (a + b)(a – b)
7q – 12 =0 of 7q + 12 = 0 → q = 1⁵∕₇ ∨ q = –1⁵∕₇
f. p² + 14p = 32
–32 = –32
p² + 14p – 32 = 0 ← Maak het rechterlid nul.
+16
– 2
(p + 16)(p – 2) = 0 ← De Product-som-methode: 16 × –2 = –32 en 16 + (–2) = 14
p + 16 = 0 of p – 2 = 0 → p = –16 ∨ p = 2
g. x² + 40 = 3x + 50
–3x – 50 = –3x – 50
x² – 3x – 10 = 0 ← Maak het rechterlid nul.
+2
–5
(x + 2)(x – 5) = 0 ← De Product-som-methode: 2 × –5 = –10 en 2 + (–5) = –3
x +2 = 0 of x – 5 = 0 → x = –2 ∨ x = 5
h. a² = 3a + 4
–3a – 4 = –3a – 4
a² –3a – 4 = 0 ← Maak het rechterlid nul.
–4
+1
(a – 4)(a + 1) = 0 ← De Product-som-methode: –4 × 1 = –4 en –4 + 1 = –3
a – 4 = 0 of a + 1 = 0 → a =4 ∨ a = –1
i. 0,1(x– 4)² = 40
×10 = ×10
(x – 4)² = 400 ← Vermenigvuldig beide zijden met 10.
– 400 = – 400
(x – 4)² – 400 = 0 ← Maak het rechterlid nul.
x² – 2∙(4)∙(x) +4² – 400 = 0
x² – 8x – 384 = 0
–24
+16
← 2⁴ = 16 – 24 + 16 = – 8
(x – 24)(x + 16) = 0 ← De Product-som-methode: –24 × 16 = –384 en –24 + 16 = –8
x – 24 = 0 of x + 16 = 0 → x = 24 ∨ x = –16
Geen opmerkingen:
Een reactie posten