Moderne Wiskunde (12.1e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 1.3 Wortelvergelijkingen

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{6 x + 27}\)

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

a

(Kwadrateren)
\(x^{2} = 6 x + 27\)

1p

○

(Oplossen)
\(x^{2} - 6 x - 27 = 0\)
\((x + 3) (x - 9) = 0\)
\(x = -3 ∨ x = 9\)

1p

○

(Controleren)
\(x = -3\) voldoet niet, \(x = 9\) voldoet.

1p

3p

b

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

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

b

(Isoleren)
\(-2 \sqrt{x} = -1\)

1p

○

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

1p

○

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

1p

4p

c

\(5 x - 2 \sqrt{x} = 7\)

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

c

(Isoleren)
\(5 x - 7 = 2 \sqrt{x}\)

1p

○

(Kwadrateren)
\((5 x - 7)^{2} = (2 \sqrt{x})^{2}\)
\(25 x^{2} - 70 x + 49 = 4 x\)

1p

○

(Oplossen)
\(25 x^{2} - 74 x + 49 = 0\)
\(D = (-74)^{2} - 4 ⋅ 25 ⋅ 49 = 576\)
\(x = {74 - \sqrt{576} \over 2 ⋅ 25} ∨ x = {74 + \sqrt{576} \over 2 ⋅ 25}\)
\(x = 1 ∨ x = 1\frac{24}{25}\)

1p

○

(Controleren)
\(x = 1\) voldoet niet, \(x = 1\frac{24}{25}\) voldoet.

1p

4p

d

\(x = \sqrt{5 x + 44} - 10\)

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

d

(Isoleren)
\(x + 10 = \sqrt{5 x + 44}\)

1p

○

(Kwadrateren)
\((x + 10)^{2} = (\sqrt{5 x + 44})^{2}\)
\(x^{2} + 20 x + 100 = 5 x + 44\)

1p

○

(Oplossen)
\(x^{2} + 15 x + 56 = 0\)
\((x + 8) (x + 7) = 0\)
\(x = -8 ∨ x = -7\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

\(6 x - 2 \sqrt{4 x - 3} = 8\)

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

○

(Isoleren)
\(6 x - 8 = 2 \sqrt{4 x - 3}\)

1p

○

(Kwadrateren)
\((6 x - 8)^{2} = (2 \sqrt{4 x - 3})^{2}\)
\(36 x^{2} - 96 x + 64 = 4 ⋅ (4 x - 3)\)
\(36 x^{2} - 96 x + 64 = 16 x - 12\)

1p

○

(Oplossen)
\(36 x^{2} - 112 x + 76 = 0\)
\(9 x^{2} - 28 x + 19 = 0\)
\(D = (-28)^{2} - 4 ⋅ 9 ⋅ 19 = 100\)
\(x = {28 - \sqrt{100} \over 2 ⋅ 9} ∨ x = {28 + \sqrt{100} \over 2 ⋅ 9}\)
\(x = 1 ∨ x = 2\frac{1}{9}\)

1p

○

(Controleren)
\(x = 1\) voldoet niet, \(x = 2\frac{1}{9}\) voldoet.

1p
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