Getal & Ruimte (12e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 5.3 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{-x + 12}\)

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

a

(Kwadrateren)
\(x^{2} = -x + 12\)

1p

○

(Oplossen)
\(1 x^{2} + 1 x + -12 = 0\)
\((x + 4) (x + -3) = 0\)
\(x = -4 ∨ x = 3\)

1p

○

(Controleren)
\(x = -4\) voldoet niet, \(x = 3\) voldoet.

1p

3p

b

\(4 + 8 \sqrt{x} = 7\)

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

b

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

1p

○

(Kwadrateren)
\((8 \sqrt{x})^{2} = 3^{2}\)
\(64 x = 9\)
\(x = \frac{9}{64}\)

1p

○

(Controleren)
\(x = \frac{9}{64}\) voldoet.

1p

4p

c

\(9 x + 3 \sqrt{x} = 6\)

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

c

(Isoleren)
\(9 x - 6 = -3 \sqrt{x}\)

1p

○

(Kwadrateren)
\((9 x - 6)^{2} = (-3 \sqrt{x})^{2}\)
\(81 x^{2} - 108 x + 36 = 9 x\)

1p

○

(Oplossen)
\(81 x^{2} + -117 x + 36 = 0\)
\(9 x^{2} + -13 x + 4 = 0\)
\(D = -13^{2} - 4 ⋅ 9 ⋅ 4 = 25\)
\(x = {13 - \sqrt{25} \over 2 ⋅ 9} ∨ x = {13 + \sqrt{25} \over 2 ⋅ 9}\)
\(x = {4 \over 9} ∨ x = 1\)

1p

○

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

1p

4p

d

\(x = \sqrt{9 x + 76} - 6\)

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

d

(Isoleren)
\(x + 6 = \sqrt{9 x + 76}\)

1p

○

(Kwadrateren)
\((x + 6)^{2} = (\sqrt{9 x + 76})^{2}\)
\(x^{2} + 12 x + 36 = 9 x + 76\)

1p

○

(Oplossen)
\(1 x^{2} + 3 x + -40 = 0\)
\((x + 8) (x + -5) = 0\)
\(x = -8 ∨ x = 5\)

1p

○

(Controleren)
\(x = -8\) voldoet niet, \(x = 5\) voldoet.

1p

opgave 2

Los exact op.

4p

\(2 x - 3 \sqrt{3 x - 5} = 2\)

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

○

(Isoleren)
\(2 x - 2 = 3 \sqrt{3 x - 5}\)

1p

○

(Kwadrateren)
\((2 x - 2)^{2} = (3 \sqrt{3 x - 5})^{2}\)
\(4 x^{2} - 8 x + 4 = 9 ⋅ (3 x - 5)\)
\(4 x^{2} - 8 x + 4 = 27 x - 45\)

1p

○

(Oplossen)
\(4 x^{2} + -35 x + 49 = 0\)
\(D = -35^{2} - 4 ⋅ 4 ⋅ 49 = 441\)
\(x = {35 - \sqrt{441} \over 2 ⋅ 4} ∨ x = {35 + \sqrt{441} \over 2 ⋅ 4}\)
\(x = {7 \over 4} ∨ x = 7\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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