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

'Wortelvergelijkingen'.

havo wiskunde B 5.2 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{-x + 30}\)

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

a

(Kwadrateren)
\(x^{2} = -x + 30\)

1p

○

(Oplossen)
\(1 x^{2} + 1 x + -30 = 0\)
\((x + 6) (x + -5) = 0\)
\(x = -6 ∨ x = 5\)

1p

○

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

1p

3p

b

\(9 - 2 \sqrt{x} = 3\)

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

b

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

1p

○

(Kwadrateren)
\((-2 \sqrt{x})^{2} = (-6)^{2}\)
\(4 x = 36\)
\(x = 9\)

1p

○

(Controleren)
\(x = 9\) voldoet.

1p

4p

c

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

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

c

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

1p

○

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

1p

○

(Oplossen)
\(64 x^{2} + -52 x + 9 = 0\)
\(D = -52^{2} - 4 ⋅ 64 ⋅ 9 = 400\)
\(x = {52 - \sqrt{400} \over 2 ⋅ 64} ∨ x = {52 + \sqrt{400} \over 2 ⋅ 64}\)
\(x = {1 \over 4} ∨ x = {9 \over 16}\)

1p

○

(Controleren)
\(x = \frac{1}{4}\) voldoet niet, \(x = \frac{9}{16}\) voldoet.

1p

4p

d

\(x = \sqrt{8 x + 57} - 6\)

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

d

(Isoleren)
\(x + 6 = \sqrt{8 x + 57}\)

1p

○

(Kwadrateren)
\((x + 6)^{2} = (\sqrt{8 x + 57})^{2}\)
\(x^{2} + 12 x + 36 = 8 x + 57\)

1p

○

(Oplossen)
\(1 x^{2} + 4 x + -21 = 0\)
\((x + 7) (x + -3) = 0\)
\(x = -7 ∨ x = 3\)

1p

○

(Controleren)
\(x = 3\) voldoet, \(x = -7\) voldoet niet.

1p

opgave 2

Los exact op.

4p

\(8 x - 7 \sqrt{4 x - 6} = 6\)

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

○

(Isoleren)
\(8 x - 6 = 7 \sqrt{4 x - 6}\)

1p

○

(Kwadrateren)
\((8 x - 6)^{2} = (7 \sqrt{4 x - 6})^{2}\)
\(64 x^{2} - 96 x + 36 = 49 ⋅ (4 x - 6)\)
\(64 x^{2} - 96 x + 36 = 196 x - 294\)

1p

○

(Oplossen)
\(64 x^{2} + -292 x + 330 = 0\)
\(32 x^{2} + -146 x + 165 = 0\)
\(D = -146^{2} - 4 ⋅ 32 ⋅ 165 = 196\)
\(x = {146 - \sqrt{196} \over 2 ⋅ 32} ∨ x = {146 + \sqrt{196} \over 2 ⋅ 32}\)
\(x = {33 \over 16} ∨ x = {5 \over 2}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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