Getal & Ruimte (13e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 5.2 Wortelfuncties

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)
\(1 x^{2} + -6 x + -27 = 0\)
\((x + 3) (x + -9) = 0\)
\(x = -3 ∨ x = 9\)

1p

○

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

1p

3p

b

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

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

b

(Isoleren)
\(-6 \sqrt{x} = -1\)

1p

○

(Kwadrateren)
\((-6 \sqrt{x})^{2} = (-1)^{2}\)
\(36 x = 1\)
\(x = \frac{1}{36}\)

1p

○

(Controleren)
\(x = \frac{1}{36}\) voldoet.

1p

4p

c

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

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

c

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

1p

○

(Kwadrateren)
\((2 x - 7)^{2} = (-5 \sqrt{x})^{2}\)
\(4 x^{2} - 28 x + 49 = 25 x\)

1p

○

(Oplossen)
\(4 x^{2} + -53 x + 49 = 0\)
\(D = -53^{2} - 4 ⋅ 4 ⋅ 49 = 2025\)
\(x = {53 - \sqrt{2025} \over 2 ⋅ 4} ∨ x = {53 + \sqrt{2025} \over 2 ⋅ 4}\)
\(x = 1 ∨ x = {49 \over 4}\)

1p

○

(Controleren)
\(x = 1\) voldoet, \(x = 12\frac{1}{4}\) voldoet niet.

1p

4p

d

\(x = \sqrt{3 x + 37} + 1\)

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

d

(Isoleren)
\(x - 1 = \sqrt{3 x + 37}\)

1p

○

(Kwadrateren)
\((x - 1)^{2} = (\sqrt{3 x + 37})^{2}\)
\(x^{2} - 2 x + 1 = 3 x + 37\)

1p

○

(Oplossen)
\(1 x^{2} + -5 x + -36 = 0\)
\((x + 4) (x + -9) = 0\)
\(x = -4 ∨ x = 9\)

1p

○

(Controleren)
\(x = -4\) voldoet niet, \(x = 9\) voldoet.

1p

opgave 2

Los exact op.

4p

\(6 x - 7 \sqrt{3 x - 6} = 6\)

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

○

(Isoleren)
\(6 x - 6 = 7 \sqrt{3 x - 6}\)

1p

○

(Kwadrateren)
\((6 x - 6)^{2} = (7 \sqrt{3 x - 6})^{2}\)
\(36 x^{2} - 72 x + 36 = 49 ⋅ (3 x - 6)\)
\(36 x^{2} - 72 x + 36 = 147 x - 294\)

1p

○

(Oplossen)
\(36 x^{2} + -219 x + 330 = 0\)
\(12 x^{2} + -73 x + 110 = 0\)
\(D = -73^{2} - 4 ⋅ 12 ⋅ 110 = 49\)
\(x = {73 - \sqrt{49} \over 2 ⋅ 12} ∨ x = {73 + \sqrt{49} \over 2 ⋅ 12}\)
\(x = {11 \over 4} ∨ x = {10 \over 3}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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