Wortelvergelijkingen
Los exact op.
3p
\(2 + 8 \sqrt{x} = 5\)
○
(Isoleren)
\(8 \sqrt{x} = 3\)
1p
○
(Kwadrateren)
\((8 \sqrt{x})^{2} = 3^{2}\)
\(64 x = 9\)
\(x = \frac{9}{64}\)
1p
○
(Controleren)
\(x = \frac{9}{64}\) voldoet.
1p
Los exact op.
3p
\(x = \sqrt{-4 x + 32}\)
○
(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
Los exact op.
4p
\(x = \sqrt{2 x + 67} - 2\)
○
(Isoleren)
\(x + 2 = \sqrt{2 x + 67}\)
1p
○
(Kwadrateren)
\((x + 2)^{2} = (\sqrt{2 x + 67})^{2}\)
\(x^{2} + 4 x + 4 = 2 x + 67\)
1p
○
(Oplossen)
\(1 x^{2} + 2 x + -63 = 0\)
\((x + 9) (x + -7) = 0\)
\(x = -9 ∨ x = 7\)
1p
○
(Controleren)
\(x = 7\) voldoet, \(x = -9\) voldoet niet.
1p
Los exact op.
4p
\(-7 x - 2 \sqrt{x} = -5\)
○
(Isoleren)
\(-7 x + 5 = 2 \sqrt{x}\)
1p
○
(Kwadrateren)
\((-7 x + 5)^{2} = (2 \sqrt{x})^{2}\)
\(49 x^{2} - 70 x + 25 = 4 x\)
1p
○
(Oplossen)
\(49 x^{2} + -74 x + 25 = 0\)
\(D = -74^{2} - 4 ⋅ 49 ⋅ 25 = 576\)
\(x = {74 - \sqrt{576} \over 2 ⋅ 49} ∨ x = {74 + \sqrt{576} \over 2 ⋅ 49}\)
\(x = {25 \over 49} ∨ x = 1\)
1p
○
(Controleren)
\(x = \frac{25}{49}\) voldoet, \(x = 1\) voldoet niet.
1p
Los exact op.
4p
\(7 x - 3 \sqrt{7 x - 9} = 7\)
○
(Isoleren)
\(7 x - 7 = 3 \sqrt{7 x - 9}\)
1p
○
(Kwadrateren)
\((7 x - 7)^{2} = (3 \sqrt{7 x - 9})^{2}\)
\(49 x^{2} - 98 x + 49 = 9 ⋅ (7 x - 9)\)
\(49 x^{2} - 98 x + 49 = 63 x - 81\)
1p
○
(Oplossen)
\(49 x^{2} + -161 x + 130 = 0\)
\(D = -161^{2} - 4 ⋅ 49 ⋅ 130 = 441\)
\(x = {161 - \sqrt{441} \over 2 ⋅ 49} ∨ x = {161 + \sqrt{441} \over 2 ⋅ 49}\)
\(x = {10 \over 7} ∨ x = {13 \over 7}\)
1p
○
(Controleren)
Beide oplossingen voldoen.
1p