VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(11+21x)=-4x^2-(75-11x)\)
2. \(7x^2-(19x-1)=3x(x-8)\)
3. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\)
4. \(x(2x+23)=18(x+1)\)
5. \(x(12x+19)=12(x+1)\)
6. \(10x^2-(16x-81)=x(x+38)\)
7. \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\)
8. \(x(x-13)=-24(x+1)\)
9. \(x(2x+23)=8(x+1)\)
10. \(x(36x+21)=-4(x+1)\)
11. \(\frac{25}{4}x=-18x^2-\frac{1}{2}\)
12. \(4x^2-(20x+3)=3x(x-6)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(11+21x)=-4x^2-(75-11x) \\ \Leftrightarrow -11-21x=-4x^2-75+11x \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
2. \(7x^2-(19x-1)=3x(x-8) \\ \Leftrightarrow 7x^2-19x+1=3x^2-24x \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}\right)=0 \color{red}{.12} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
4. \(x(2x+23)=18(x+1) \\ \Leftrightarrow 2x^2+23x=18x+18 \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
5. \(x(12x+19)=12(x+1) \\ \Leftrightarrow 12x^2+19x=12x+12 \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(10x^2-(16x-81)=x(x+38) \\ \Leftrightarrow 10x^2-16x+81=x^2+38x \\ \Leftrightarrow 9x^2-54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}=0\\ \Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2+\frac{1}{5}x+\frac{3}{5}\right)=0 \color{red}{.60} \\ \Leftrightarrow x^2+12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
8. \(x(x-13)=-24(x+1) \\ \Leftrightarrow x^2-13x=-24x-24 \\ \Leftrightarrow x^2+11x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.24 & &\\ & = 121-96 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt25}{2.1} & & = \frac{-11+\sqrt25}{2.1} \\ & = \frac{-16}{2} & & = \frac{-6}{2} \\ & = -8 & & = -3 \\ \\ V &= \Big\{ -8 ; -3 \Big\} & &\end{align} \\ -----------------\)
9. \(x(2x+23)=8(x+1) \\ \Leftrightarrow 2x^2+23x=8x+8 \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(x(36x+21)=-4(x+1) \\ \Leftrightarrow 36x^2+21x=-4x-4 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{25}{4}x=-18x^2-\frac{1}{2} \\ \Leftrightarrow 18x^2+\frac{25}{4}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(18x^2+\frac{25}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(4x^2-(20x+3)=3x(x-6) \\ \Leftrightarrow 4x^2-20x-3=3x^2-18x \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)