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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-5x+2)(-2x-3)-x(-2x-16)=-18\)
2. \(-x=-\frac{1}{3}x^2+\frac{28}{3}\)
3. \((2x+2)(3x-5)-x(5x-19)=80\)
4. \(\frac{1}{5}x^2-\frac{1}{5}x+\frac{1}{20}=0\)
5. \(x(48x+22)=-3(x+1)\)
6. \(-(8-20x)=-x^2-(-32-17x)\)
7. \(49x^2-(10x+3)=x(x-17)\)
8. \(x(16x-9)=-(x+1)\)
9. \(x(9x+13)=11(x-11)\)
10. \(x(9x+28)=10(x-10)\)
11. \(x(x+53)=60(x+1)\)
12. \(\frac{3}{2}x^2-\frac{23}{3}x+\frac{50}{3}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-5x+2)(-2x-3)-x(-2x-16)=-18\\ \Leftrightarrow 10x^2+15x-4x-6 +2x^2+16x+18=0 \\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(-x=-\frac{1}{3}x^2+\frac{28}{3} \\ \Leftrightarrow \frac{1}{3}x^2-x-\frac{28}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-x-\frac{28}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
3. \((2x+2)(3x-5)-x(5x-19)=80\\ \Leftrightarrow 6x^2-10x+6x-10 -5x^2+19x-80=0 \\ \Leftrightarrow x^2-x-90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-90) & &\\ & = 1+360 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt361}{2.1} & & = \frac{-(-1)+\sqrt361}{2.1} \\ & = \frac{-18}{2} & & = \frac{20}{2} \\ & = -9 & & = 10 \\ \\ V &= \Big\{ -9 ; 10 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{5}x^2-\frac{1}{5}x+\frac{1}{20}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2-\frac{1}{5}x+\frac{1}{20}\right)=0 \color{red}{.20} \\ \Leftrightarrow 16x^2-16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.16} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(x(48x+22)=-3(x+1) \\ \Leftrightarrow 48x^2+22x=-3x-3 \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
6. \(-(8-20x)=-x^2-(-32-17x) \\ \Leftrightarrow -8+20x=-x^2+32+17x \\ \Leftrightarrow x^2+3x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-40) & &\\ & = 9+160 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt169}{2.1} & & = \frac{-3+\sqrt169}{2.1} \\ & = \frac{-16}{2} & & = \frac{10}{2} \\ & = -8 & & = 5 \\ \\ V &= \Big\{ -8 ; 5 \Big\} & &\end{align} \\ -----------------\)
7. \(49x^2-(10x+3)=x(x-17) \\ \Leftrightarrow 49x^2-10x-3=x^2-17x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
8. \(x(16x-9)=-(x+1) \\ \Leftrightarrow 16x^2-9x=-x-1 \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(x(9x+13)=11(x-11) \\ \Leftrightarrow 9x^2+13x=11x-121 \\ \Leftrightarrow 9x^2+2x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+2x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.9.121 & &\\ & = 4-4356 & & \\ & = -4352 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(x(9x+28)=10(x-10) \\ \Leftrightarrow 9x^2+28x=10x-100 \\ \Leftrightarrow 9x^2+18x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+18x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.9.100 & &\\ & = 324-3600 & & \\ & = -3276 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(x(x+53)=60(x+1) \\ \Leftrightarrow x^2+53x=60x+60 \\ \Leftrightarrow x^2-7x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-60) & &\\ & = 49+240 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt289}{2.1} & & = \frac{-(-7)+\sqrt289}{2.1} \\ & = \frac{-10}{2} & & = \frac{24}{2} \\ & = -5 & & = 12 \\ \\ V &= \Big\{ -5 ; 12 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{3}{2}x^2-\frac{23}{3}x+\frac{50}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{3}{2}x^2-\frac{23}{3}x+\frac{50}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 9x^2-46x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-46x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-46)^2-4.9.100 & &\\ & = 2116-3600 & & \\ & = -1484 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)