VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{19}{5}x=-\frac{1}{5}x^2-\frac{84}{5}\)
2. \(9x^2-(18x+80)=8x(x-2)\)
3. \(\frac{25}{12}x=-\frac{1}{3}x^2-3\)
4. \(18x^2-(2x-1)=2x(x-6)\)
5. \(2x^2-(18x-60)=x(x-1)\)
6. \(-(6-17x)=-x^2-(18-9x)\)
7. \(-(11-37x)=-x^2-(91-19x)\)
8. \(2x^2-(19x+42)=x(x-18)\)
9. \(37x^2-(14x+4)=x(x-21)\)
10. \((4x+2)(2x-1)-x(7x-26)=-102\)
11. \((5x+4)(4x+5)-x(-28x+38)=23\)
12. \(11x^2-(20x+4)=7x(x-5)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{19}{5}x=-\frac{1}{5}x^2-\frac{84}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{19}{5}x+\frac{84}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{19}{5}x+\frac{84}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt25}{2.1} & & = \frac{-19+\sqrt25}{2.1} \\ & = \frac{-24}{2} & & = \frac{-14}{2} \\ & = -12 & & = -7 \\ \\ V &= \Big\{ -12 ; -7 \Big\} & &\end{align} \\ -----------------\)
2. \(9x^2-(18x+80)=8x(x-2) \\ \Leftrightarrow 9x^2-18x-80=8x^2-16x \\ \Leftrightarrow x^2-2x-80=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-80) & &\\ & = 4+320 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt324}{2.1} & & = \frac{-(-2)+\sqrt324}{2.1} \\ & = \frac{-16}{2} & & = \frac{20}{2} \\ & = -8 & & = 10 \\ \\ V &= \Big\{ -8 ; 10 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{25}{12}x=-\frac{1}{3}x^2-3 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{25}{12}x+3=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+\frac{25}{12}x+3\right)=0 \color{red}{.12} \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(18x^2-(2x-1)=2x(x-6) \\ \Leftrightarrow 18x^2-2x+1=2x^2-12x \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
5. \(2x^2-(18x-60)=x(x-1) \\ \Leftrightarrow 2x^2-18x+60=x^2-x \\ \Leftrightarrow x^2-17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-17)-\sqrt49}{2.1} & & = \frac{-(-17)+\sqrt49}{2.1} \\ & = \frac{10}{2} & & = \frac{24}{2} \\ & = 5 & & = 12 \\ \\ V &= \Big\{ 5 ; 12 \Big\} & &\end{align} \\ -----------------\)
6. \(-(6-17x)=-x^2-(18-9x) \\ \Leftrightarrow -6+17x=-x^2-18+9x \\ \Leftrightarrow x^2+8x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.12 & &\\ & = 64-48 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt16}{2.1} & & = \frac{-8+\sqrt16}{2.1} \\ & = \frac{-12}{2} & & = \frac{-4}{2} \\ & = -6 & & = -2 \\ \\ V &= \Big\{ -6 ; -2 \Big\} & &\end{align} \\ -----------------\)
7. \(-(11-37x)=-x^2-(91-19x) \\ \Leftrightarrow -11+37x=-x^2-91+19x \\ \Leftrightarrow x^2+18x+80=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+80=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.80 & &\\ & = 324-320 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-18-\sqrt4}{2.1} & & = \frac{-18+\sqrt4}{2.1} \\ & = \frac{-20}{2} & & = \frac{-16}{2} \\ & = -10 & & = -8 \\ \\ V &= \Big\{ -10 ; -8 \Big\} & &\end{align} \\ -----------------\)
8. \(2x^2-(19x+42)=x(x-18) \\ \Leftrightarrow 2x^2-19x-42=x^2-18x \\ \Leftrightarrow x^2-x-42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-42=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-42) & &\\ & = 1+168 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt169}{2.1} & & = \frac{-(-1)+\sqrt169}{2.1} \\ & = \frac{-12}{2} & & = \frac{14}{2} \\ & = -6 & & = 7 \\ \\ V &= \Big\{ -6 ; 7 \Big\} & &\end{align} \\ -----------------\)
9. \(37x^2-(14x+4)=x(x-21) \\ \Leftrightarrow 37x^2-14x-4=x^2-21x \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
10. \((4x+2)(2x-1)-x(7x-26)=-102\\ \Leftrightarrow 8x^2-4x+4x-2 -7x^2+26x+102=0 \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
11. \((5x+4)(4x+5)-x(-28x+38)=23\\ \Leftrightarrow 20x^2+25x+16x+20 +28x^2-38x-23=0 \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(11x^2-(20x+4)=7x(x-5) \\ \Leftrightarrow 11x^2-20x-4=7x^2-35x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)