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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{2}x^2-\frac{11}{2}x+9=0\)
2. \(\frac{1}{2}x=-\frac{1}{20}x^2-\frac{5}{4}\)
3. \(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}=0\)
4. \((-x+4)(2x+2)-x(-11x+1)=12\)
5. \(29x^2-(17x+2)=11x(x-2)\)
6. \(\frac{1}{2}x=-\frac{1}{3}x^2+\frac{1}{3}\)
7. \((3x-3)(3x-4)-x(8x-6)=19\)
8. \(65x^2-(9x-3)=17x(x-2)\)
9. \(\frac{1}{8}x^2+\frac{11}{4}x+15=0\)
10. \(10x^2-(6x-4)=x(x+6)\)
11. \((-3x-5)(3x-4)-x(-10x+12)=-80\)
12. \(8x^2+\frac{17}{2}x+\frac{1}{2}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{2}x^2-\frac{11}{2}x+9=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{11}{2}x+9\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-11x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.18 & &\\ & = 121-72 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt49}{2.1} & & = \frac{-(-11)+\sqrt49}{2.1} \\ & = \frac{4}{2} & & = \frac{18}{2} \\ & = 2 & & = 9 \\ \\ V &= \Big\{ 2 ; 9 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{2}x=-\frac{1}{20}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{20}x^2+\frac{1}{2}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{20}x^2+\frac{1}{2}x+\frac{5}{4}\right)=0 \color{red}{.20} \\ \Leftrightarrow x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-10}{2.1} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow x^2-3x-54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt225}{2.1} & & = \frac{-(-3)+\sqrt225}{2.1} \\ & = \frac{-12}{2} & & = \frac{18}{2} \\ & = -6 & & = 9 \\ \\ V &= \Big\{ -6 ; 9 \Big\} & &\end{align} \\ -----------------\)
4. \((-x+4)(2x+2)-x(-11x+1)=12\\ \Leftrightarrow -2x^2-2x+8x+8 +11x^2-x-12=0 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
5. \(29x^2-(17x+2)=11x(x-2) \\ \Leftrightarrow 29x^2-17x-2=11x^2-22x \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{1}{2}x=-\frac{1}{3}x^2+\frac{1}{3} \\ \Leftrightarrow \frac{1}{3}x^2+\frac{1}{2}x-\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{1}{2}x-\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
7. \((3x-3)(3x-4)-x(8x-6)=19\\ \Leftrightarrow 9x^2-12x-9x+12 -8x^2+6x-19=0 \\ \Leftrightarrow x^2+6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt64}{2.1} & & = \frac{-6+\sqrt64}{2.1} \\ & = \frac{-14}{2} & & = \frac{2}{2} \\ & = -7 & & = 1 \\ \\ V &= \Big\{ -7 ; 1 \Big\} & &\end{align} \\ -----------------\)
8. \(65x^2-(9x-3)=17x(x-2) \\ \Leftrightarrow 65x^2-9x+3=17x^2-34x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{8}x^2+\frac{11}{4}x+15=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{11}{4}x+15\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-22-\sqrt4}{2.1} & & = \frac{-22+\sqrt4}{2.1} \\ & = \frac{-24}{2} & & = \frac{-20}{2} \\ & = -12 & & = -10 \\ \\ V &= \Big\{ -12 ; -10 \Big\} & &\end{align} \\ -----------------\)
10. \(10x^2-(6x-4)=x(x+6) \\ \Leftrightarrow 10x^2-6x+4=x^2+6x \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
11. \((-3x-5)(3x-4)-x(-10x+12)=-80\\ \Leftrightarrow -9x^2+12x-15x+20 +10x^2-12x+80=0 \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
12. \(8x^2+\frac{17}{2}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(8x^2+\frac{17}{2}x+\frac{1}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)