Getal & Ruimte (13e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 5.2 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{x + 56}\)

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

a

(Kwadrateren)
\(x^{2} = x + 56\)

1p

○

(Oplossen)
\(x^{2} - x - 56 = 0\)
\((x + 7) (x - 8) = 0\)
\(x = -7 ∨ x = 8\)

1p

○

(Controleren)
\(x = -7\) voldoet niet, \(x = 8\) voldoet.

1p

3p

b

\(6 + 8 \sqrt{x} = 7\)

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

b

(Isoleren)
\(8 \sqrt{x} = 1\)

1p

○

(Kwadrateren)
\((8 \sqrt{x})^{2} = 1^{2}\)
\(64 x = 1\)
\(x = \frac{1}{64}\)

1p

○

(Controleren)
\(x = \frac{1}{64}\) voldoet.

1p

4p

c

\(8 x + 3 \sqrt{x} = 5\)

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

c

(Isoleren)
\(8 x - 5 = -3 \sqrt{x}\)

1p

○

(Kwadrateren)
\((8 x - 5)^{2} = (-3 \sqrt{x})^{2}\)
\(64 x^{2} - 80 x + 25 = 9 x\)

1p

○

(Oplossen)
\(64 x^{2} - 89 x + 25 = 0\)
\(D = (-89)^{2} - 4 ⋅ 64 ⋅ 25 = 1\,521\)
\(x = {89 - \sqrt{1\,521} \over 2 ⋅ 64} ∨ x = {89 + \sqrt{1\,521} \over 2 ⋅ 64}\)
\(x = \frac{25}{64} ∨ x = 1\)

1p

○

(Controleren)
\(x = \frac{25}{64}\) voldoet, \(x = 1\) voldoet niet.

1p

4p

d

\(x = \sqrt{6 x + 24} - 4\)

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

d

(Isoleren)
\(x + 4 = \sqrt{6 x + 24}\)

1p

○

(Kwadrateren)
\((x + 4)^{2} = (\sqrt{6 x + 24})^{2}\)
\(x^{2} + 8 x + 16 = 6 x + 24\)

1p

○

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

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

\(8 x - 4 \sqrt{3 x - 3} = 8\)

Wortel (5)
008r - Wortelvergelijkingen - basis - 594ms - dynamic variables

○

(Isoleren)
\(8 x - 8 = 4 \sqrt{3 x - 3}\)

1p

○

(Kwadrateren)
\((8 x - 8)^{2} = (4 \sqrt{3 x - 3})^{2}\)
\(64 x^{2} - 128 x + 64 = 16 ⋅ (3 x - 3)\)
\(64 x^{2} - 128 x + 64 = 48 x - 48\)

1p

○

(Oplossen)
\(64 x^{2} - 176 x + 112 = 0\)
\(4 x^{2} - 11 x + 7 = 0\)
\(D = (-11)^{2} - 4 ⋅ 4 ⋅ 7 = 9\)
\(x = {11 - \sqrt{9} \over 2 ⋅ 4} ∨ x = {11 + \sqrt{9} \over 2 ⋅ 4}\)
\(x = 1 ∨ x = 1\frac{3}{4}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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