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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(6x+33)=8(x-3)\)
2. \(21x^2-(20x+12)=9x(x-3)\)
3. \(-\frac{1}{2}x=-\frac{1}{20}x^2-\frac{5}{4}\)
4. \(-(8-31x)=-x^2-(38-18x)\)
5. \(17x^2-(14x-100)=x(x+66)\)
6. \(\frac{25}{8}x=-x^2-\frac{9}{4}\)
7. \(x(9x-95)=-121(x+1)\)
8. \(-(10-6x)=-x^2-(-100-7x)\)
9. \(14x^2-(18x+33)=13x(x-2)\)
10. \(x(x-15)=3(x-27)\)
11. \(x(72x+23)=-2(x+1)\)
12. \(-\frac{43}{5}x=-\frac{8}{5}x^2-\frac{121}{10}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(6x+33)=8(x-3) \\ \Leftrightarrow 6x^2+33x=8x-24 \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(21x^2-(20x+12)=9x(x-3) \\ \Leftrightarrow 21x^2-20x-12=9x^2-27x \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(-\frac{1}{2}x=-\frac{1}{20}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{20}x^2-\frac{1}{2}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{20}x^2-\frac{1}{2}x+\frac{5}{4}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2-40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.4} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
4. \(-(8-31x)=-x^2-(38-18x) \\ \Leftrightarrow -8+31x=-x^2-38+18x \\ \Leftrightarrow x^2+13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt49}{2.1} & & = \frac{-13+\sqrt49}{2.1} \\ & = \frac{-20}{2} & & = \frac{-6}{2} \\ & = -10 & & = -3 \\ \\ V &= \Big\{ -10 ; -3 \Big\} & &\end{align} \\ -----------------\)
5. \(17x^2-(14x-100)=x(x+66) \\ \Leftrightarrow 17x^2-14x+100=x^2+66x \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{25}{8}x=-x^2-\frac{9}{4} \\ \Leftrightarrow x^2+\frac{25}{8}x+\frac{9}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{25}{8}x+\frac{9}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(x(9x-95)=-121(x+1) \\ \Leftrightarrow 9x^2-95x=-121x-121 \\ \Leftrightarrow 9x^2+26x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+26x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (26)^2-4.9.121 & &\\ & = 676-4356 & & \\ & = -3680 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-(10-6x)=-x^2-(-100-7x) \\ \Leftrightarrow -10+6x=-x^2+100+7x \\ \Leftrightarrow x^2-x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-110) & &\\ & = 1+440 & & \\ & = 441 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt441}{2.1} & & = \frac{-(-1)+\sqrt441}{2.1} \\ & = \frac{-20}{2} & & = \frac{22}{2} \\ & = -10 & & = 11 \\ \\ V &= \Big\{ -10 ; 11 \Big\} & &\end{align} \\ -----------------\)
9. \(14x^2-(18x+33)=13x(x-2) \\ \Leftrightarrow 14x^2-18x-33=13x^2-26x \\ \Leftrightarrow x^2+8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt196}{2.1} & & = \frac{-8+\sqrt196}{2.1} \\ & = \frac{-22}{2} & & = \frac{6}{2} \\ & = -11 & & = 3 \\ \\ V &= \Big\{ -11 ; 3 \Big\} & &\end{align} \\ -----------------\)
10. \(x(x-15)=3(x-27) \\ \Leftrightarrow x^2-15x=3x-81 \\ \Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-18)}{2.1} & & \\ & = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
11. \(x(72x+23)=-2(x+1) \\ \Leftrightarrow 72x^2+23x=-2x-2 \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(-\frac{43}{5}x=-\frac{8}{5}x^2-\frac{121}{10} \\ \Leftrightarrow \frac{8}{5}x^2-\frac{43}{5}x+\frac{121}{10}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{8}{5}x^2-\frac{43}{5}x+\frac{121}{10}\right)=0 \color{red}{.10} \\ \Leftrightarrow 16x^2-86x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-86x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-86)^2-4.16.121 & &\\ & = 7396-7744 & & \\ & = -348 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)