Moderne Wiskunde (12.1e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 1.3 Wortelvergelijkingen

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{4 x + 45}\)

Wortel (2)
008n - Wortelvergelijkingen - basis - 0ms - dynamic variables

a

(Kwadrateren)
\(x^{2} = 4 x + 45\)

1p

○

(Oplossen)
\(1 x^{2} + -4 x + -45 = 0\)
\((x + 5) (x + -9) = 0\)
\(x = -5 ∨ x = 9\)

1p

○

(Controleren)
\(x = -5\) voldoet niet, \(x = 9\) voldoet.

1p

3p

b

\(3 + 4 \sqrt{x} = 5\)

Wortel (1)
008o - Wortelvergelijkingen - basis - 1ms - dynamic variables

b

(Isoleren)
\(4 \sqrt{x} = 2\)

1p

○

(Kwadrateren)
\((4 \sqrt{x})^{2} = 2^{2}\)
\(16 x = 4\)
\(x = \frac{1}{4}\)

1p

○

(Controleren)
\(x = \frac{1}{4}\) voldoet.

1p

4p

c

\(-7 x - 4 \sqrt{x} = -3\)

Wortel (4)
008p - Wortelvergelijkingen - basis - 5ms - dynamic variables

c

(Isoleren)
\(-7 x + 3 = 4 \sqrt{x}\)

1p

○

(Kwadrateren)
\((-7 x + 3)^{2} = (4 \sqrt{x})^{2}\)
\(49 x^{2} - 42 x + 9 = 16 x\)

1p

○

(Oplossen)
\(49 x^{2} + -58 x + 9 = 0\)
\(D = -58^{2} - 4 ⋅ 49 ⋅ 9 = 1600\)
\(x = {58 - \sqrt{1600} \over 2 ⋅ 49} ∨ x = {58 + \sqrt{1600} \over 2 ⋅ 49}\)
\(x = {9 \over 49} ∨ x = 1\)

1p

○

(Controleren)
\(x = \frac{9}{49}\) voldoet, \(x = 1\) voldoet niet.

1p

4p

d

\(x = \sqrt{6 x - 23} + 3\)

Wortel (3)
008q - Wortelvergelijkingen - basis - 0ms - dynamic variables

d

(Isoleren)
\(x - 3 = \sqrt{6 x - 23}\)

1p

○

(Kwadrateren)
\((x - 3)^{2} = (\sqrt{6 x - 23})^{2}\)
\(x^{2} - 6 x + 9 = 6 x - 23\)

1p

○

(Oplossen)
\(1 x^{2} + -12 x + 32 = 0\)
\((x + -4) (x + -8) = 0\)
\(x = 4 ∨ x = 8\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

\(4 x - 5 \sqrt{2 x - 4} = 8\)

Wortel (5)
008r - Wortelvergelijkingen - basis - 489ms - dynamic variables

○

(Isoleren)
\(4 x - 8 = 5 \sqrt{2 x - 4}\)

1p

○

(Kwadrateren)
\((4 x - 8)^{2} = (5 \sqrt{2 x - 4})^{2}\)
\(16 x^{2} - 64 x + 64 = 25 ⋅ (2 x - 4)\)
\(16 x^{2} - 64 x + 64 = 50 x - 100\)

1p

○

(Oplossen)
\(16 x^{2} + -114 x + 164 = 0\)
\(8 x^{2} + -57 x + 82 = 0\)
\(D = -57^{2} - 4 ⋅ 8 ⋅ 82 = 625\)
\(x = {57 - \sqrt{625} \over 2 ⋅ 8} ∨ x = {57 + \sqrt{625} \over 2 ⋅ 8}\)
\(x = 2 ∨ x = {41 \over 8}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
"