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

'Wortelvergelijkingen'.

havo wiskunde B 5.2 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{2 x + 24}\)

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

a

(Kwadrateren)
\(x^{2} = 2 x + 24\)

1p

○

(Oplossen)
\(1 x^{2} + -2 x + -24 = 0\)
\((x + 4) (x + -6) = 0\)
\(x = -4 ∨ x = 6\)

1p

○

(Controleren)
\(x = -4\) voldoet niet, \(x = 6\) voldoet.

1p

3p

b

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

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

\(9 x + 3 \sqrt{x} = 2\)

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

c

(Isoleren)
\(9 x - 2 = -3 \sqrt{x}\)

1p

○

(Kwadrateren)
\((9 x - 2)^{2} = (-3 \sqrt{x})^{2}\)
\(81 x^{2} - 36 x + 4 = 9 x\)

1p

○

(Oplossen)
\(81 x^{2} + -45 x + 4 = 0\)
\(D = -45^{2} - 4 ⋅ 81 ⋅ 4 = 729\)
\(x = {45 - \sqrt{729} \over 2 ⋅ 81} ∨ x = {45 + \sqrt{729} \over 2 ⋅ 81}\)
\(x = {1 \over 9} ∨ x = {4 \over 9}\)

1p

○

(Controleren)
\(x = \frac{1}{9}\) voldoet, \(x = \frac{4}{9}\) voldoet niet.

1p

4p

d

\(x = \sqrt{9 x + 46} - 4\)

Wortel (3)
008q - Wortelvergelijkingen - basis - 1ms - dynamic variables

d

(Isoleren)
\(x + 4 = \sqrt{9 x + 46}\)

1p

○

(Kwadrateren)
\((x + 4)^{2} = (\sqrt{9 x + 46})^{2}\)
\(x^{2} + 8 x + 16 = 9 x + 46\)

1p

○

(Oplossen)
\(1 x^{2} + -1 x + -30 = 0\)
\((x + 5) (x + -6) = 0\)
\(x = -5 ∨ x = 6\)

1p

○

(Controleren)
\(x = -5\) voldoet niet, \(x = 6\) voldoet.

1p

opgave 2

Los exact op.

4p

\(8 x - 3 \sqrt{6 x - 2} = 2\)

Wortel (5)
008r - Wortelvergelijkingen - basis - 511ms - dynamic variables

○

(Isoleren)
\(8 x - 2 = 3 \sqrt{6 x - 2}\)

1p

○

(Kwadrateren)
\((8 x - 2)^{2} = (3 \sqrt{6 x - 2})^{2}\)
\(64 x^{2} - 32 x + 4 = 9 ⋅ (6 x - 2)\)
\(64 x^{2} - 32 x + 4 = 54 x - 18\)

1p

○

(Oplossen)
\(64 x^{2} + -86 x + 22 = 0\)
\(32 x^{2} + -43 x + 11 = 0\)
\(D = -43^{2} - 4 ⋅ 32 ⋅ 11 = 441\)
\(x = {43 - \sqrt{441} \over 2 ⋅ 32} ∨ x = {43 + \sqrt{441} \over 2 ⋅ 32}\)
\(x = {11 \over 32} ∨ x = 1\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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