Wortelvergelijkingen
Los exact op.
3p
\(4 + 2 \sqrt{x} = 8\)
○
(Isoleren)
\(2 \sqrt{x} = 4\)
1p
○
(Kwadrateren)
\((2 \sqrt{x})^{2} = 4^{2}\)
\(4 x = 16\)
\(x = 4\)
1p
○
(Controleren)
\(x = 4\) voldoet.
1p
Los exact op.
3p
\(x = \sqrt{-x + 56}\)
○
(Kwadrateren)
\(x^{2} = -x + 56\)
1p
○
(Oplossen)
\(1 x^{2} + 1 x + -56 = 0\)
\((x + 8) (x + -7) = 0\)
\(x = -8 ∨ x = 7\)
1p
○
(Controleren)
\(x = -8\) voldoet niet, \(x = 7\) voldoet.
1p
Los exact op.
4p
\(x = \sqrt{9 x + 73} - 7\)
○
(Isoleren)
\(x + 7 = \sqrt{9 x + 73}\)
1p
○
(Kwadrateren)
\((x + 7)^{2} = (\sqrt{9 x + 73})^{2}\)
\(x^{2} + 14 x + 49 = 9 x + 73\)
1p
○
(Oplossen)
\(1 x^{2} + 5 x + -24 = 0\)
\((x + 8) (x + -3) = 0\)
\(x = -8 ∨ x = 3\)
1p
○
(Controleren)
\(x = 3\) voldoet, \(x = -8\) voldoet niet.
1p
Los exact op.
4p
\(2 x - 9 \sqrt{x} = -9\)
○
(Isoleren)
\(2 x + 9 = 9 \sqrt{x}\)
1p
○
(Kwadrateren)
\((2 x + 9)^{2} = (9 \sqrt{x})^{2}\)
\(4 x^{2} + 36 x + 81 = 81 x\)
1p
○
(Oplossen)
\(4 x^{2} + -45 x + 81 = 0\)
\(D = -45^{2} - 4 ⋅ 4 ⋅ 81 = 729\)
\(x = {45 - \sqrt{729} \over 2 ⋅ 4} ∨ x = {45 + \sqrt{729} \over 2 ⋅ 4}\)
\(x = {9 \over 4} ∨ x = 9\)
1p
○
(Controleren)
Beide oplossingen voldoen.
1p
Los exact op.
4p
\(4 x - 3 \sqrt{6 x - 5} = 2\)
○
(Isoleren)
\(4 x - 2 = 3 \sqrt{6 x - 5}\)
1p
○
(Kwadrateren)
\((4 x - 2)^{2} = (3 \sqrt{6 x - 5})^{2}\)
\(16 x^{2} - 16 x + 4 = 9 ⋅ (6 x - 5)\)
\(16 x^{2} - 16 x + 4 = 54 x - 45\)
1p
○
(Oplossen)
\(16 x^{2} + -70 x + 49 = 0\)
\(D = -70^{2} - 4 ⋅ 16 ⋅ 49 = 1764\)
\(x = {70 - \sqrt{1764} \over 2 ⋅ 16} ∨ x = {70 + \sqrt{1764} \over 2 ⋅ 16}\)
\(x = {7 \over 8} ∨ x = {7 \over 2}\)
1p
○
(Controleren)
Beide oplossingen voldoen.
1p