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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(65x^2-(9x-3)=17x(x-2)\)
2. \(17x^2-(18x-9)=x(x+6)\)
3. \(x(3x+29)=4(x-12)\)
4. \(\frac{1}{3}x^2-6x+\frac{77}{3}=0\)
5. \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}=0\)
6. \(x(x+22)=16(x+1)\)
7. \(x(16x-77)=11(x-11)\)
8. \((-4x-4)(-2x+3)-x(7x-7)=-72\)
9. \(9x^2-(9x-18)=x(x-34)\)
10. \((x+1)(3x+4)-x(-13x-88)=-140\)
11. \((-x-5)(-4x+4)-x(-68x-31)=-18\)
12. \(3x^2+\frac{5}{3}x-\frac{4}{3}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(65x^2-(9x-3)=17x(x-2) \\ \Leftrightarrow 65x^2-9x+3=17x^2-34x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(17x^2-(18x-9)=x(x+6) \\ \Leftrightarrow 17x^2-18x+9=x^2+6x \\ \Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.16} & & \\ & = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(x(3x+29)=4(x-12) \\ \Leftrightarrow 3x^2+29x=4x-48 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{3}x^2-6x+\frac{77}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-6x+\frac{77}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-18x+77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.77 & &\\ & = 324-308 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-18)-\sqrt16}{2.1} & & = \frac{-(-18)+\sqrt16}{2.1} \\ & = \frac{14}{2} & & = \frac{22}{2} \\ & = 7 & & = 11 \\ \\ V &= \Big\{ 7 ; 11 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2+3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-18) & &\\ & = 9+72 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt81}{2.1} & & = \frac{-3+\sqrt81}{2.1} \\ & = \frac{-12}{2} & & = \frac{6}{2} \\ & = -6 & & = 3 \\ \\ V &= \Big\{ -6 ; 3 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+22)=16(x+1) \\ \Leftrightarrow x^2+22x=16x+16 \\ \Leftrightarrow x^2+6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-16) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.1} & & = \frac{-6+\sqrt100}{2.1} \\ & = \frac{-16}{2} & & = \frac{4}{2} \\ & = -8 & & = 2 \\ \\ V &= \Big\{ -8 ; 2 \Big\} & &\end{align} \\ -----------------\)
7. \(x(16x-77)=11(x-11) \\ \Leftrightarrow 16x^2-77x=11x-121 \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
8. \((-4x-4)(-2x+3)-x(7x-7)=-72\\ \Leftrightarrow 8x^2-12x+8x-12 -7x^2+7x+72=0 \\ \Leftrightarrow x^2-17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt49}{2.1} & & = \frac{-(-17)+\sqrt49}{2.1} \\ & = \frac{10}{2} & & = \frac{24}{2} \\ & = 5 & & = 12 \\ \\ V &= \Big\{ 5 ; 12 \Big\} & &\end{align} \\ -----------------\)
9. \(9x^2-(9x-18)=x(x-34) \\ \Leftrightarrow 9x^2-9x+18=x^2-34x \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
10. \((x+1)(3x+4)-x(-13x-88)=-140\\ \Leftrightarrow 3x^2+4x+3x+4 +13x^2+88x+140=0 \\ \Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-96}{2.16} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
11. \((-x-5)(-4x+4)-x(-68x-31)=-18\\ \Leftrightarrow 4x^2-4x+20x-20 +68x^2+31x+18=0 \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(3x^2+\frac{5}{3}x-\frac{4}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(3x^2+\frac{5}{3}x-\frac{4}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)