Antwoorde 1 / Answers 1
Die vergelykings is in die vorm : ax2 + bx + c = 0
Die formule word gegee deur
──────
─ b ± √ b2 ─ 4ac
x = ──────────────
2a
Skryf elke keer die waardes van a, b en c neer,
vul dit in die formule in en bereken die antwoord :
The equations are given by : ax2 + bx + c = 0
The formula is given by
──────
─ b ± √ b2 ─ 4ac
x = ──────────────
2a
Each time write down the values of a, b and c,
substitute into the formula and calculate the answer :
1.1 x
2 + 3x + 2 = 0
a = 1; b = 3; c = 2
Die formule / The formula
──────
─ b ± √ b
2 ─ 4ac
x = ──────────────
2a
Vul die waardes in / Substitute the values :
─────────
─ 3
+ √ 3
2 ─ 4(1)(2)
x = ───────────────── = ─ 1
2(1)
OF / OR
─────────
─ 3
─ √ 3
2 ─ 4(1)(2)
x = ──────────────── = ─ 2
2(1)
x = ─ 1 OF / OR x = ─ 2
[ V / Q 1.1 ]
1.3 x
2 + 4x ─ 21 = 0
a = 1; b = 4 c = ─21
x = 3 OF / OR x = ─7
[ 1.3 ]
1.5 x
2 + 8x + 15 = 0
a = 1; b = 8 c = 15
x = ─3 OF / OR x = ─5
[ 1.5 ]
1.7 x
2 + 6x − 2 = 0
a = 1; b = 6 c = ─2
x = 0,32 OF / OR x = ─6,32
[ 1.7 ]
1.9 2x
2 + 9x − 24 = 0
a = 2; b = 9 c = ─ 24
x = 1,88 OF / OR x = ─ 6,38
[ 1.9 ]
1.11 7y
2 + 3y − 6 = 0
a = 7; b = 3 c = ─ 6
y = 0,74 OF / OR y = ─ 1,16
[ 1.11 ]
1.13 3y
2 − 2y − 4 = 0
a = 3; b = ─ 2 c = ─ 4
y = 1,54 OF / OR y = ─ 0,87
[ 1.13 ]
1.15 3y
2 + 5y + 1 = 0
a = 3; b = 5 c = 1
y = ─ 0,23 OF / OR y = ─ 1,43
[ 1.15 ]
1.17 24 − 10y − y
2 = 0
a = ─ 1; b = ─ 10 c = 24
y = ─ 12 OF / OR y = 2
[ 1.17 ]
1.19 7 − 5a − 4a
2 = 0
a = ─ 4; b = ─ 5 c = 7
a = ─ 2,09 OF / OR a = 0,84
[ 1.19 ]
1.2 x
2 ─ 3x + 2 = 0
a = 1; b = ─3 c = 2
Die formule / The formula
──────
─ b ± √ b
2 ─ 4ac
x = ──────────────
2a
Vul die waardes in / Substitute the values :
─────────
─ (─3)
+ √ (─3)
2 ─ 4(1)(2)
x = ───────────────── = 2
2(1)
OF / OR
─────────
─ (─3)
─ √ (─3)
2 ─ 4(1)(2)
x = ──────────────── = 1
2(1)
x = 2 OF / OR x = 1
[ V / Q 1.2 ]
1.4 x
2 ─ 10x + 24 = 0
a = 1; b = ─ 10 c = 24
x = 6 OF / OR x = 4
[ 1.4 ]
1.6 x
2 ─ 2x − 4 = 0
a = 1; b = ─ 2 c = ─ 4
x = 3,24 OF / OR x = ─ 1,24
[ 1.6 ]
1.8 x
2 + 22x − 15 = 0
a = 1; b = 22 c = ─ 15
x = 0,66 OF / OR x = ─ 22,66
[ 1.8 ]
1.10 5x
2 ─ 8x + 2 = 0
a = 5; b = ─ 8 c = 2
x = 1,29 OF / OR x = 0,31
[ 1.10 ]
1.12 y
2 + 5y − 7 = 0
a = 1; b = 5 c = ─ 7
y = 1,14 OF / OR y = ─ 6,14
[ 1.12 ]
1.14 2y
2 + y − 5 = 0
a = 2; b = 1 c = ─ 5
y = 1,35 OF / OR y = ─ 1,85
[ 1.14 ]
1.16 2 − 3x − x
2 = 0
a = ─ 1; b = ─ 3 c = 2
x = ─ 3,56 OF / OR x = 0,56
[ 1.16 ]
1.18 15 + 2y − y
2 = 0
a = ─ 1; b = 2 c = 15
y = ─ 3 OF / OR y = 5
[ 1.18 ]
1.20 7 − 3a − 4a
2 = 0
a = ─ 4; b = ─ 3 c = 7
a = ─ 1,75 OF / OR a = 1
[ 1.20 ]
Antwoorde 2 / Answers 2
Onthou om die periode in jaar te gee, d.w.s.
verander die maande na jare
23
bv. 23 maande = ─── jaar
12
2.1 a = 39 244,26
[ 2.1 ]
2.3 c = 6 132,19
[ 2.3 ]
2.5 e = 12
[ 2.5 ]
2.7 g = 8,75
[ 2.7 ]
Remember that the period must be given in years,
therefore change the months into years,
23
e.g. 23 months = ─── years
12
2.2 b = 309 635,94
[ 2.2 ]
2.4 d = 6 722,48
[ 2.4 ]
2.6 f = 23,5
[ 2.6 ]
2.8 h = 10,6
[ 2.8 ]
Antwoorde 3 / Answers 3
3.1 v
2 = u
2 + 2as
─ 2as = ─ v
2 + u
2
s moet die onderwerp word en dus moet s aan die linkerkant wees. /
s must be the subject of the formula and therefore s has to be on the
lefthand side.
2as = v
2 ─ u
2
verander die tekens / change the signs.
v
2 ─ u
2
s = ─────────
kry s alleen - deel met 2a / get s by itself - divide by 2a.
2a
[ 3.1 ]
3.2 v
2 = u
2 + 2as
─ 2as = ─ v
2 + u
2
a moet die onderwerp word en dus moet a aan die linkerkant wees. /
a must be the subject of the formula and therefore a has to be on the
lefthand side.
2as = v
2 ─ u
2
verander die tekens / change the signs.
v
2 ─ u
2
a = ─────────
kry a alleen - deel met 2s / get a by itself - divide by 2s.
2s
[ 3.2 ]
3.3 v
2 = u
2 + 2as
─ u
2 = ─ v
2 + 2as
u moet die onderwerp word en dus moet u aan die linkerkant wees. /
u must be the subject of the formula and therefore u has to be on the
lefthand side.
u
2 = v
2 ─ 2as
verander die tekens / change the signs.
────────
u = √ v
2 ─ 2as
[ 3.3 ]
Antwoorde 4 / Answers 4
Antwoorde 5 / Answers 5
5.1 s = ut + ½at
2
─ ut = ─ s + ½at
2
u moet die onderwerp word en dus moet u aan die linkerkant wees. /
u must be the subject of the formula and therefore u has to be on the
lefthand side.
ut = s ─ ½at
2
verander die tekens / change the signs.
2ut = 2s ─ at
2
X 2 om die ½ te verwyder / X 2 to remove the ½
2s ─ at
2
u = ───────
kry u alleen - deel met 2t / get u by itself - divide by 2t.
2t
[ 5.1 ]
5.2 s = ut + ½at
2
2s = 2ut + at
2
X2 - verwyder die breuk / X 2 - remove the fraction
─ at
2 = 2ut ─ 2s
a moet die onderwerp word en dus moet a aan die linkerkant wees. /
a must be the subject of the formula and therefore a has to be on the
lefthand side.
at
2 = ─ 2ut + 2s
verander die tekens / change the signs.
2(s ─ ut)
a = ───────
kry a alleen - deel met t
2 / get a by itself - divide by t
2.
t
2
[ 5.2 ]
5.3 s = ut + ½at
2
2s = 2ut + at
2
X 2 - verwyder die breuk / X 2 - remove the fraction
─ at
2 ─ 2ut + 2s = 0
maak 'n kwadratiese vergelyking in t want t moet die onderwerp word. /
t must be the subject of the formula and therefore make a
quadratic equation in t.
at
2 + 2ut ─ 2s = 0
verander die tekens / change the signs.
Dit is nou 'n kwadratiese vergelyking - / It is now a quadratic equation -
los op met formule 1.3 solve by using formula 3.
a = a; b = 2u ; c = ─ 2s
──────
─ b ± √ b
2 ─ 4ac
t = ──────────────
2a
───────────
─ (2u) ± √ (2u)
2 ─ 4a(─ 2s)
= ──────────────────── vul waardes in / substitute values
2(a)
───────
─ 2u ± √ 4u
2 + 8as
= ──────────────── vereenvoudig / simplify
2a
──────
─ 2u ± 2√ u
2 + 2as
= ──────────────── haal √4 as gemene faktor uit / remove √4 as common factor
2a
──────
─ u ± √ u
2 + 2as
= ──────────────── deel met 2 / divide by 2
a
[ 5.3 ]
Antwoorde 6 / Answers 6
Antwoorde 7 / Answers 7
n
7.1 S = ─ [2a + (n ─ 1)d]
2
2S = n[2a + (n ─ 1)d]
X 2 om die ½ te verwyder / X2 to remove the ½
2S = 2an + n
2d ─ nd
verwyder die hakies / remove the brackets
─ 2an = ─ 2S + n
2d ─ nd
a moet die onderwerp word en dus moet a aan die linkerkant wees. /
a must be the subject of the formula and therefore a has to be on the
lefthand side.
2an = 2S ─ n
2d + nd
verander die tekens. / change signs.
2an = 2S ─ nd(n ─ 1)
vereenvoudig. / simplify.
2S ─ n(n ─ 1)d
a = ─────────────
kry a alleen - deel met 2n. / get a by itself - divide by 2n.
2n
[ 7.1 ]
n
7.2 S = ─ [2a + (n ─ 1)d]
2
2S = n[2a + (n ─ 1)d]
X 2 om die ½ te verwyder / X2 to remove the ½
2S = 2an + n(n ─ 1)d
verwyder die hakies / remove the brackets
─ n(n ─ 1)d = 2an ─ 2S
d moet die onderwerp word en dus moet d aan die linkerkant wees. /
d must be the subject of the formula and therefore d has to be on the
lefthand side.
n(n ─ 1)d = 2S ─ 2an
verander die tekens. / change signs.
2(S ─ an)
d = ────────
kry d alleen - deel met n(n - 1). / get d alone - divide by n(n - 1).
n(n ─ 1)
[ 7.2 ]
Antwoorde 8 / Answers 8
Antwoorde 9 / Answers 9
l
9.1 T = 2π √ ───
g
l
T
2 = 4π
2 ───
kwadreer om √ te verwyder / square to remove the √
g
T
2g = 4π
2l
X g om breuke te verwyder / X g to remove the fraction
T
2g
l = ───
kry l alleen - deel met 4π
2. / get l by itself - divide by 4π
2.
4π
2
[ 9.1 ]
l
9.2 T = 2π √ ───
g
l
T
2 = 4π
2 ───
kwadreer om √ te verwyder / square to remove the √
g
T
2g = 4π
2l
X g om breuke te verwyder / X g to remove the fraction
4π
2l
g = ────
kry g alleen - deel met T
2. / get g by itself - divide by 4T
2.
T
2
[ 9.2 ]
Antwoorde 10 / Answers 10