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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((5x-4)(-4x-5)-x(-21x-25)=-80\)
2. \(17x^2-(9x-121)=x(x+79)\)
3. \(\frac{7}{5}x=-\frac{1}{10}x^2-\frac{12}{5}\)
4. \(-(8+9x)=-x^2-(44-3x)\)
5. \(-(8-82x)=-16x^2-(72-18x)\)
6. \(-(10-8x)=-2x^2-(8-5x)\)
7. \(-\frac{5}{2}x=-\frac{1}{2}x^2+12\)
8. \(17x^2-(17x+1)=x(x-23)\)
9. \((-x+3)(3x+2)-x(-12x-1)=10\)
10. \(-(8-16x)=-x^2-(-2-19x)\)
11. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5}\)
12. \(-\frac{2}{3}x=-\frac{1}{33}x^2-\frac{11}{3}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((5x-4)(-4x-5)-x(-21x-25)=-80\\ \Leftrightarrow -20x^2-25x+16x+20 +21x^2+25x+80=0 \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
2. \(17x^2-(9x-121)=x(x+79) \\ \Leftrightarrow 17x^2-9x+121=x^2+79x \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{7}{5}x=-\frac{1}{10}x^2-\frac{12}{5} \\ \Leftrightarrow \frac{1}{10}x^2+\frac{7}{5}x+\frac{12}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2+\frac{7}{5}x+\frac{12}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow x^2+14x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.24 & &\\ & = 196-96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt100}{2.1} & & = \frac{-14+\sqrt100}{2.1} \\ & = \frac{-24}{2} & & = \frac{-4}{2} \\ & = -12 & & = -2 \\ \\ V &= \Big\{ -12 ; -2 \Big\} & &\end{align} \\ -----------------\)
4. \(-(8+9x)=-x^2-(44-3x) \\ \Leftrightarrow -8-9x=-x^2-44+3x \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
5. \(-(8-82x)=-16x^2-(72-18x) \\ \Leftrightarrow -8+82x=-16x^2-72+18x \\ \Leftrightarrow 16x^2+64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+64x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (64)^2-4.16.64 & &\\ & = 4096-4096 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-64}{2.16} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
6. \(-(10-8x)=-2x^2-(8-5x) \\ \Leftrightarrow -10+8x=-2x^2-8+5x \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(-\frac{5}{2}x=-\frac{1}{2}x^2+12 \\ \Leftrightarrow \frac{1}{2}x^2-\frac{5}{2}x-12=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{5}{2}x-12\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt121}{2.1} & & = \frac{-(-5)+\sqrt121}{2.1} \\ & = \frac{-6}{2} & & = \frac{16}{2} \\ & = -3 & & = 8 \\ \\ V &= \Big\{ -3 ; 8 \Big\} & &\end{align} \\ -----------------\)
8. \(17x^2-(17x+1)=x(x-23) \\ \Leftrightarrow 17x^2-17x-1=x^2-23x \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \((-x+3)(3x+2)-x(-12x-1)=10\\ \Leftrightarrow -3x^2-2x+9x+6 +12x^2+x-10=0 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
10. \(-(8-16x)=-x^2-(-2-19x) \\ \Leftrightarrow -8+16x=-x^2+2+19x \\ \Leftrightarrow x^2-3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt49}{2.1} & & = \frac{-(-3)+\sqrt49}{2.1} \\ & = \frac{-4}{2} & & = \frac{10}{2} \\ & = -2 & & = 5 \\ \\ V &= \Big\{ -2 ; 5 \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{7}{15}x=-\frac{16}{5}x^2+\frac{1}{5} \\ \Leftrightarrow \frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{16}{5}x^2+\frac{7}{15}x-\frac{1}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(-\frac{2}{3}x=-\frac{1}{33}x^2-\frac{11}{3} \\ \Leftrightarrow \frac{1}{33}x^2-\frac{2}{3}x+\frac{11}{3}=0 \\ \Leftrightarrow \color{red}{33.} \left(\frac{1}{33}x^2-\frac{2}{3}x+\frac{11}{3}\right)=0 \color{red}{.33} \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)