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

'Wortelvergelijkingen'.

havo wiskunde B 5.3 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{x + 20}\)

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

a

(Kwadrateren)
\(x^{2} = x + 20\)

1p

○

(Oplossen)
\(x^{2} - x - 20 = 0\)
\((x + 4) (x - 5) = 0\)
\(x = -4 ∨ x = 5\)

1p

○

(Controleren)
\(x = -4\) voldoet niet, \(x = 5\) voldoet.

1p

3p

b

\(4 + 8 \sqrt{x} = 6\)

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

b

(Isoleren)
\(8 \sqrt{x} = 2\)

1p

○

(Kwadrateren)
\((8 \sqrt{x})^{2} = 2^{2}\)
\(64 x = 4\)
\(x = \frac{1}{16}\)

1p

○

(Controleren)
\(x = \frac{1}{16}\) voldoet.

1p

4p

c

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

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

c

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

1p

○

(Kwadrateren)
\((4 x - 2)^{2} = (2 \sqrt{x})^{2}\)
\(16 x^{2} - 16 x + 4 = 4 x\)

1p

○

(Oplossen)
\(16 x^{2} - 20 x + 4 = 0\)
\(4 x^{2} - 5 x + 1 = 0\)
\(D = (-5)^{2} - 4 ⋅ 4 ⋅ 1 = 9\)
\(x = {5 - \sqrt{9} \over 2 ⋅ 4} ∨ x = {5 + \sqrt{9} \over 2 ⋅ 4}\)
\(x = \frac{1}{4} ∨ x = 1\)

1p

○

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

1p

4p

d

\(x = \sqrt{3 x + 31} - 1\)

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

d

(Isoleren)
\(x + 1 = \sqrt{3 x + 31}\)

1p

○

(Kwadrateren)
\((x + 1)^{2} = (\sqrt{3 x + 31})^{2}\)
\(x^{2} + 2 x + 1 = 3 x + 31\)

1p

○

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

1p

○

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

1p

opgave 2

Los exact op.

4p

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

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

○

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

1p

○

(Kwadrateren)
\((3 x - 8)^{2} = (2 \sqrt{2 x - 3})^{2}\)
\(9 x^{2} - 48 x + 64 = 4 ⋅ (2 x - 3)\)
\(9 x^{2} - 48 x + 64 = 8 x - 12\)

1p

○

(Oplossen)
\(9 x^{2} - 56 x + 76 = 0\)
\(D = (-56)^{2} - 4 ⋅ 9 ⋅ 76 = 400\)
\(x = {56 - \sqrt{400} \over 2 ⋅ 9} ∨ x = {56 + \sqrt{400} \over 2 ⋅ 9}\)
\(x = 2 ∨ x = 4\frac{2}{9}\)

1p

○

(Controleren)
\(x = 2\) voldoet niet, \(x = 4\frac{2}{9}\) voldoet.

1p
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