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

'Wortelvergelijkingen'.

havo wiskunde B 5.2 Wortelfuncties

Wortelvergelijkingen (5)

opgave 1

Los exact op.

3p

a

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

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

a

(Kwadrateren)
\(x^{2} = -2 x + 8\)

1p

○

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

1p

○

(Controleren)
\(x = -4\) voldoet niet, \(x = 2\) voldoet.

1p

3p

b

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

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

b

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

1p

○

(Kwadrateren)
\((-2 \sqrt{x})^{2} = (-2)^{2}\)
\(4 x = 4\)
\(x = 1\)

1p

○

(Controleren)
\(x = 1\) voldoet.

1p

4p

c

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

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

c

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

1p

○

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

1p

○

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

1p

○

(Controleren)
\(x = \frac{4}{49}\) voldoet niet, \(x = 1\) voldoet.

1p

4p

d

\(x = \sqrt{7 x + 44} - 2\)

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

d

(Isoleren)
\(x + 2 = \sqrt{7 x + 44}\)

1p

○

(Kwadrateren)
\((x + 2)^{2} = (\sqrt{7 x + 44})^{2}\)
\(x^{2} + 4 x + 4 = 7 x + 44\)

1p

○

(Oplossen)
\(1 x^{2} + -3 x + -40 = 0\)
\((x + 5) (x + -8) = 0\)
\(x = -5 ∨ x = 8\)

1p

○

(Controleren)
\(x = -5\) voldoet niet, \(x = 8\) voldoet.

1p

opgave 2

Los exact op.

4p

\(6 x - 2 \sqrt{2 x - 2} = 7\)

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

○

(Isoleren)
\(6 x - 7 = 2 \sqrt{2 x - 2}\)

1p

○

(Kwadrateren)
\((6 x - 7)^{2} = (2 \sqrt{2 x - 2})^{2}\)
\(36 x^{2} - 84 x + 49 = 4 ⋅ (2 x - 2)\)
\(36 x^{2} - 84 x + 49 = 8 x - 8\)

1p

○

(Oplossen)
\(36 x^{2} + -92 x + 57 = 0\)
\(D = -92^{2} - 4 ⋅ 36 ⋅ 57 = 256\)
\(x = {92 - \sqrt{256} \over 2 ⋅ 36} ∨ x = {92 + \sqrt{256} \over 2 ⋅ 36}\)
\(x = {19 \over 18} ∨ x = {3 \over 2}\)

1p

○

(Controleren)
\(x = 1\frac{1}{18}\) voldoet niet, \(x = 1\frac{1}{2}\) voldoet.

1p
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