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

1p

○

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

1p

3p

b

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

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

b

(Isoleren)
\(7 \sqrt{x} = 4\)

1p

○

(Kwadrateren)
\((7 \sqrt{x})^{2} = 4^{2}\)
\(49 x = 16\)
\(x = \frac{16}{49}\)

1p

○

(Controleren)
\(x = \frac{16}{49}\) voldoet.

1p

4p

c

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

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

c

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

1p

○

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

1p

○

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

1p

○

(Controleren)
\(x = \frac{25}{49}\) voldoet, \(x = 1\) voldoet niet.

1p

4p

d

\(x = \sqrt{8 x + 16} - 2\)

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

d

(Isoleren)
\(x + 2 = \sqrt{8 x + 16}\)

1p

○

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

1p

○

(Oplossen)
\(1 x^{2} + -4 x + -12 = 0\)
\((x + 2) (x + -6) = 0\)
\(x = -2 ∨ x = 6\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

\(5 x - 2 \sqrt{8 x - 7} = 3\)

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

○

(Isoleren)
\(5 x - 3 = 2 \sqrt{8 x - 7}\)

1p

○

(Kwadrateren)
\((5 x - 3)^{2} = (2 \sqrt{8 x - 7})^{2}\)
\(25 x^{2} - 30 x + 9 = 4 ⋅ (8 x - 7)\)
\(25 x^{2} - 30 x + 9 = 32 x - 28\)

1p

○

(Oplossen)
\(25 x^{2} + -62 x + 37 = 0\)
\(D = -62^{2} - 4 ⋅ 25 ⋅ 37 = 144\)
\(x = {62 - \sqrt{144} \over 2 ⋅ 25} ∨ x = {62 + \sqrt{144} \over 2 ⋅ 25}\)
\(x = 1 ∨ x = {37 \over 25}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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