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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x-3)=3(x-3)\)
2. \(x(16x-78)=2(x-50)\)
3. \((3x+2)(5x+1)-x(-33x-2)=5\)
4. \(19x^2-(18x+2)=11x(x-3)\)
5. \(\frac{7}{2}x=-\frac{1}{2}x^2+22\)
6. \(x(x+23)=6(x-12)\)
7. \(-(10-27x)=-3x^2-(22-14x)\)
8. \(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}=0\)
9. \(3x^2+\frac{25}{6}x+\frac{4}{3}=0\)
10. \(-(10+12x)=-x^2-(58-2x)\)
11. \(x(x+13)=15(x+1)\)
12. \(\frac{1}{2}x^2+\frac{25}{4}x+18=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x-3)=3(x-3) \\ \Leftrightarrow x^2-3x=3x-9 \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
2. \(x(16x-78)=2(x-50) \\ \Leftrightarrow 16x^2-78x=2x-100 \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
3. \((3x+2)(5x+1)-x(-33x-2)=5\\ \Leftrightarrow 15x^2+3x+10x+2 +33x^2+2x-5=0 \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
4. \(19x^2-(18x+2)=11x(x-3) \\ \Leftrightarrow 19x^2-18x-2=11x^2-33x \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{7}{2}x=-\frac{1}{2}x^2+22 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{2}x-22=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{7}{2}x-22\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt225}{2.1} & & = \frac{-7+\sqrt225}{2.1} \\ & = \frac{-22}{2} & & = \frac{8}{2} \\ & = -11 & & = 4 \\ \\ V &= \Big\{ -11 ; 4 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+23)=6(x-12) \\ \Leftrightarrow x^2+23x=6x-72 \\ \Leftrightarrow x^2+17x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.72 & &\\ & = 289-288 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt1}{2.1} & & = \frac{-17+\sqrt1}{2.1} \\ & = \frac{-18}{2} & & = \frac{-16}{2} \\ & = -9 & & = -8 \\ \\ V &= \Big\{ -9 ; -8 \Big\} & &\end{align} \\ -----------------\)
7. \(-(10-27x)=-3x^2-(22-14x) \\ \Leftrightarrow -10+27x=-3x^2-22+14x \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{13}{10}x+\frac{9}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
9. \(3x^2+\frac{25}{6}x+\frac{4}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(3x^2+\frac{25}{6}x+\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(-(10+12x)=-x^2-(58-2x) \\ \Leftrightarrow -10-12x=-x^2-58+2x \\ \Leftrightarrow x^2-14x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.48 & &\\ & = 196-192 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt4}{2.1} & & = \frac{-(-14)+\sqrt4}{2.1} \\ & = \frac{12}{2} & & = \frac{16}{2} \\ & = 6 & & = 8 \\ \\ V &= \Big\{ 6 ; 8 \Big\} & &\end{align} \\ -----------------\)
11. \(x(x+13)=15(x+1) \\ \Leftrightarrow x^2+13x=15x+15 \\ \Leftrightarrow x^2-2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt64}{2.1} & & = \frac{-(-2)+\sqrt64}{2.1} \\ & = \frac{-6}{2} & & = \frac{10}{2} \\ & = -3 & & = 5 \\ \\ V &= \Big\{ -3 ; 5 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{1}{2}x^2+\frac{25}{4}x+18=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{25}{4}x+18\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)