Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-15x+44=0\)
2. \(x^2+17x+60=0\)
3. \(x^2+4x+67=-12x+3\)
4. \(x^2-3x-25=-12x-3\)
5. \(16x^2+87x+112=7x+12\)
6. \(18x^2+5x-2=0\)
7. \(x^2-2x-6=3x-12\)
8. \(48x^2+7x-3=0\)
9. \(x^2-3x-54=0\)
10. \(x^2+14x+45=0\)
11. \(x^2-x+32=11x+5\)
12. \(x^2+12x+36=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt49}{2.1} & & = \frac{-(-15)+\sqrt49}{2.1} \\ & = \frac{8}{2} & & = \frac{22}{2} \\ & = 4 & & = 11 \\ \\ V &= \Big\{ 4 ; 11 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
3. \(x^2+4x+67=-12x+3\\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-3x-25=-12x-3\\ \Leftrightarrow x^2+9x-22=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-22=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-22) & &\\ & = 81+88 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt169}{2.1} & & = \frac{-9+\sqrt169}{2.1} \\ & = \frac{-22}{2} & & = \frac{4}{2} \\ & = -11 & & = 2 \\ \\ V &= \Big\{ -11 ; 2 \Big\} & &\end{align} \\ -----------------\)
5. \(16x^2+87x+112=7x+12\\ \Leftrightarrow 16x^2+80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-80}{2.16} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
7. \(x^2-2x-6=3x-12\\ \Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\ & = \frac{4}{2} & & = \frac{6}{2} \\ & = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt225}{2.1} & & = \frac{-(-3)+\sqrt225}{2.1} \\ & = \frac{-12}{2} & & = \frac{18}{2} \\ & = -6 & & = 9 \\ \\ V &= \Big\{ -6 ; 9 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.45 & &\\ & = 196-180 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt16}{2.1} & & = \frac{-14+\sqrt16}{2.1} \\ & = \frac{-18}{2} & & = \frac{-10}{2} \\ & = -9 & & = -5 \\ \\ V &= \Big\{ -9 ; -5 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2-x+32=11x+5\\ \Leftrightarrow x^2-12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.27 & &\\ & = 144-108 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt36}{2.1} & & = \frac{-(-12)+\sqrt36}{2.1} \\ & = \frac{6}{2} & & = \frac{18}{2} \\ & = 3 & & = 9 \\ \\ V &= \Big\{ 3 ; 9 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)