Moderne Wiskunde (12.1e editie) - havo wiskunde B

'Wortelvergelijkingen'.

havo wiskunde B 1.3 Wortelvergelijkingen

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

\(x = \sqrt{-4 x + 32}\)

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

a

(Kwadrateren)
\(x^{2} = -4 x + 32\)

1p

○

(Oplossen)
\(1 x^{2} + 4 x + -32 = 0\)
\((x + 8) (x + -4) = 0\)
\(x = -8 ∨ x = 4\)

1p

○

(Controleren)
\(x = -8\) voldoet niet, \(x = 4\) voldoet.

1p

3p

b

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

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

b

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

1p

○

(Kwadrateren)
\((-7 \sqrt{x})^{2} = (-3)^{2}\)
\(49 x = 9\)
\(x = \frac{9}{49}\)

1p

○

(Controleren)
\(x = \frac{9}{49}\) voldoet.

1p

4p

c

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

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

c

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

1p

○

(Kwadrateren)
\((-2 x - 5)^{2} = (-7 \sqrt{x})^{2}\)
\(4 x^{2} + 20 x + 25 = 49 x\)

1p

○

(Oplossen)
\(4 x^{2} + -29 x + 25 = 0\)
\(D = -29^{2} - 4 ⋅ 4 ⋅ 25 = 441\)
\(x = {29 - \sqrt{441} \over 2 ⋅ 4} ∨ x = {29 + \sqrt{441} \over 2 ⋅ 4}\)
\(x = 1 ∨ x = {25 \over 4}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

4p

d

\(x = \sqrt{8 x + 24} - 3\)

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

d

(Isoleren)
\(x + 3 = \sqrt{8 x + 24}\)

1p

○

(Kwadrateren)
\((x + 3)^{2} = (\sqrt{8 x + 24})^{2}\)
\(x^{2} + 6 x + 9 = 8 x + 24\)

1p

○

(Oplossen)
\(1 x^{2} + -2 x + -15 = 0\)
\((x + 3) (x + -5) = 0\)
\(x = -3 ∨ x = 5\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p

opgave 2

Los exact op.

4p

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

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

○

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

1p

○

(Kwadrateren)
\((2 x - 3)^{2} = (5 \sqrt{2 x - 7})^{2}\)
\(4 x^{2} - 12 x + 9 = 25 ⋅ (2 x - 7)\)
\(4 x^{2} - 12 x + 9 = 50 x - 175\)

1p

○

(Oplossen)
\(4 x^{2} + -62 x + 184 = 0\)
\(2 x^{2} + -31 x + 92 = 0\)
\(D = -31^{2} - 4 ⋅ 2 ⋅ 92 = 225\)
\(x = {31 - \sqrt{225} \over 2 ⋅ 2} ∨ x = {31 + \sqrt{225} \over 2 ⋅ 2}\)
\(x = 4 ∨ x = {23 \over 2}\)

1p

○

(Controleren)
Beide oplossingen voldoen.

1p
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