Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(16x^2-96x+144=0\)
2. \(4x^2+20x+40=-5x+4\)
3. \(9x^2-54x+81=0\)
4. \(x^2+17x+60=0\)
5. \(x^2-5x-84=0\)
6. \(4x^2+12x+9=0\)
7. \(x^2-2x+144=0\)
8. \(x^2-12x+11=0\)
9. \(16x^2-54x+45=2x-4\)
10. \(x^2-5x-14=0\)
11. \(x^2-3x+0=7x-9\)
12. \(4x^2+14x-2=-x+2\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-96)}{2.16} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
2. \(4x^2+20x+40=-5x+4\\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt361}{2.1} & & = \frac{-(-5)+\sqrt361}{2.1} \\ & = \frac{-14}{2} & & = \frac{24}{2} \\ & = -7 & & = 12 \\ \\ V &= \Big\{ -7 ; 12 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.144 & &\\ & = 4-576 & & \\ & = -572 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.11 & &\\ & = 144-44 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt100}{2.1} & & = \frac{-(-12)+\sqrt100}{2.1} \\ & = \frac{2}{2} & & = \frac{22}{2} \\ & = 1 & & = 11 \\ \\ V &= \Big\{ 1 ; 11 \Big\} & &\end{align} \\ -----------------\)
9. \(16x^2-54x+45=2x-4\\ \Leftrightarrow 16x^2-56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-56)}{2.16} & & \\ & = \frac{7}{4} & & \\V &= \Big\{ \frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt81}{2.1} & & = \frac{-(-5)+\sqrt81}{2.1} \\ & = \frac{-4}{2} & & = \frac{14}{2} \\ & = -2 & & = 7 \\ \\ V &= \Big\{ -2 ; 7 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-3x+0=7x-9\\ \Leftrightarrow x^2-10x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.9 & &\\ & = 100-36 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt64}{2.1} & & = \frac{-(-10)+\sqrt64}{2.1} \\ & = \frac{2}{2} & & = \frac{18}{2} \\ & = 1 & & = 9 \\ \\ V &= \Big\{ 1 ; 9 \Big\} & &\end{align} \\ -----------------\)
12. \(4x^2+14x-2=-x+2\\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)