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

'Wortelvergelijkingen'.

havo wiskunde B 5.3 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{7 x + 18}\)

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

a

(Kwadrateren)
\(x^{2} = 7 x + 18\)

1p

○

(Oplossen)
\(1 x^{2} + -7 x + -18 = 0\)
\((x + 2) (x + -9) = 0\)
\(x = -2 ∨ x = 9\)

1p

○

(Controleren)
\(x = -2\) voldoet niet, \(x = 9\) voldoet.

1p

3p

b

\(9 - 7 \sqrt{x} = 4\)

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

b

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

1p

○

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

1p

○

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

1p

4p

c

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

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

c

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

1p

○

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

1p

○

(Oplossen)
\(4 x^{2} + -17 x + 4 = 0\)
\(D = -17^{2} - 4 ⋅ 4 ⋅ 4 = 225\)
\(x = {17 - \sqrt{225} \over 2 ⋅ 4} ∨ x = {17 + \sqrt{225} \over 2 ⋅ 4}\)
\(x = {1 \over 4} ∨ x = 4\)

1p

○

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

1p

4p

d

\(x = \sqrt{8 x + 65} - 10\)

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

d

(Isoleren)
\(x + 10 = \sqrt{8 x + 65}\)

1p

○

(Kwadrateren)
\((x + 10)^{2} = (\sqrt{8 x + 65})^{2}\)
\(x^{2} + 20 x + 100 = 8 x + 65\)

1p

○

(Oplossen)
\(1 x^{2} + 12 x + 35 = 0\)
\((x + 7) (x + 5) = 0\)
\(x = -7 ∨ x = -5\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

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

Wortel (5)
008r - Wortelvergelijkingen - basis - 560ms - dynamic variables

○

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

1p

○

(Kwadrateren)
\((5 x - 5)^{2} = (2 \sqrt{2 x - 2})^{2}\)
\(25 x^{2} - 50 x + 25 = 4 ⋅ (2 x - 2)\)
\(25 x^{2} - 50 x + 25 = 8 x - 8\)

1p

○

(Oplossen)
\(25 x^{2} + -58 x + 33 = 0\)
\(D = -58^{2} - 4 ⋅ 25 ⋅ 33 = 64\)
\(x = {58 - \sqrt{64} \over 2 ⋅ 25} ∨ x = {58 + \sqrt{64} \over 2 ⋅ 25}\)
\(x = 1 ∨ x = {33 \over 25}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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