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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{11}{3}x=-\frac{1}{6}x^2-24\)
2. \(x(4x+3)=-(x+1)\)
3. \(-(13-6x)=-x^2-(8-10x)\)
4. \((-x-1)(3x+1)-x(-7x+42)=-122\)
5. \(\frac{9}{16}x^2-\frac{3}{2}x+1=0\)
6. \((2x+2)(-5x-4)-x(-11x-17)=34\)
7. \(-\frac{1}{5}x=-\frac{1}{100}x^2-1\)
8. \(\frac{1}{4}x^2-\frac{5}{4}x+\frac{81}{16}=0\)
9. \(x(2x+31)=6(x-12)\)
10. \(-\frac{3}{2}x=-\frac{9}{8}x^2-\frac{1}{2}\)
11. \(23x^2-(8x+8)=5x(x-3)\)
12. \(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{11}{3}x=-\frac{1}{6}x^2-24 \\ \Leftrightarrow \frac{1}{6}x^2+\frac{11}{3}x+24=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+\frac{11}{3}x+24\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+22x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.144 & &\\ & = 484-576 & & \\ & = -92 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(x(4x+3)=-(x+1) \\ \Leftrightarrow 4x^2+3x=-x-1 \\ \Leftrightarrow 4x^2+4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(-(13-6x)=-x^2-(8-10x) \\ \Leftrightarrow -13+6x=-x^2-8+10x \\ \Leftrightarrow x^2-4x-5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-5) & &\\ & = 16+20 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt36}{2.1} & & = \frac{-(-4)+\sqrt36}{2.1} \\ & = \frac{-2}{2} & & = \frac{10}{2} \\ & = -1 & & = 5 \\ \\ V &= \Big\{ -1 ; 5 \Big\} & &\end{align} \\ -----------------\)
4. \((-x-1)(3x+1)-x(-7x+42)=-122\\ \Leftrightarrow -3x^2-x-3x-1 +7x^2-42x+122=0 \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{9}{16}x^2-\frac{3}{2}x+1=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{16}x^2-\frac{3}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
6. \((2x+2)(-5x-4)-x(-11x-17)=34\\ \Leftrightarrow -10x^2-8x-10x-8 +11x^2+17x-34=0 \\ \Leftrightarrow x^2+x-42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-42) & &\\ & = 1+168 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt169}{2.1} & & = \frac{-1+\sqrt169}{2.1} \\ & = \frac{-14}{2} & & = \frac{12}{2} \\ & = -7 & & = 6 \\ \\ V &= \Big\{ -7 ; 6 \Big\} & &\end{align} \\ -----------------\)
7. \(-\frac{1}{5}x=-\frac{1}{100}x^2-1 \\ \Leftrightarrow \frac{1}{100}x^2-\frac{1}{5}x+1=0 \\ \Leftrightarrow \color{red}{100.} \left(\frac{1}{100}x^2-\frac{1}{5}x+1\right)=0 \color{red}{.100} \\ \Leftrightarrow x^2-20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.1} & & \\ & = 10 & & \\V &= \Big\{ 10 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{4}x^2-\frac{5}{4}x+\frac{81}{16}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2-\frac{5}{4}x+\frac{81}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-20x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-20x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.4.81 & &\\ & = 400-1296 & & \\ & = -896 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(x(2x+31)=6(x-12) \\ \Leftrightarrow 2x^2+31x=6x-72 \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(-\frac{3}{2}x=-\frac{9}{8}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{9}{8}x^2-\frac{3}{2}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{8}x^2-\frac{3}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(23x^2-(8x+8)=5x(x-3) \\ \Leftrightarrow 23x^2-8x-8=5x^2-15x \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)