VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{7}{8}x=-\frac{1}{4}x^2+9\)
2. \(-(15-45x)=-4x^2-(96-9x)\)
3. \(\frac{5}{2}x=-\frac{36}{5}x^2-\frac{1}{5}\)
4. \(\frac{1}{2}x^2+\frac{3}{4}x-\frac{1}{2}=0\)
5. \((-3x+3)(4x+4)-x(-16x-44)=-109\)
6. \(24x^2-(17x-2)=6x(x-5)\)
7. \(x(x+81)=77(x+1)\)
8. \(5x^2-(18x-16)=x(x-2)\)
9. \(10x^2-(4x-100)=x(x+56)\)
10. \(x(24x+28)=3(x-2)\)
11. \(\frac{4}{3}x^2+\frac{13}{3}x+3=0\)
12. \(\frac{16}{55}x^2-\frac{8}{5}x+\frac{11}{5}=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{7}{8}x=-\frac{1}{4}x^2+9 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{7}{8}x-9=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{7}{8}x-9\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(-(15-45x)=-4x^2-(96-9x) \\ \Leftrightarrow -15+45x=-4x^2-96+9x \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{5}{2}x=-\frac{36}{5}x^2-\frac{1}{5} \\ \Leftrightarrow \frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{2}x^2+\frac{3}{4}x-\frac{1}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{3}{4}x-\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \((-3x+3)(4x+4)-x(-16x-44)=-109\\ \Leftrightarrow -12x^2-12x+12x+12 +16x^2+44x+109=0 \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(24x^2-(17x-2)=6x(x-5) \\ \Leftrightarrow 24x^2-17x+2=6x^2-30x \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
7. \(x(x+81)=77(x+1) \\ \Leftrightarrow x^2+81x=77x+77 \\ \Leftrightarrow x^2+4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt324}{2.1} & & = \frac{-4+\sqrt324}{2.1} \\ & = \frac{-22}{2} & & = \frac{14}{2} \\ & = -11 & & = 7 \\ \\ V &= \Big\{ -11 ; 7 \Big\} & &\end{align} \\ -----------------\)
8. \(5x^2-(18x-16)=x(x-2) \\ \Leftrightarrow 5x^2-18x+16=x^2-2x \\ \Leftrightarrow 4x^2-16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.4} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
9. \(10x^2-(4x-100)=x(x+56) \\ \Leftrightarrow 10x^2-4x+100=x^2+56x \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(x(24x+28)=3(x-2) \\ \Leftrightarrow 24x^2+28x=3x-6 \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{4}{3}x^2+\frac{13}{3}x+3=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{13}{3}x+3\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{16}{55}x^2-\frac{8}{5}x+\frac{11}{5}=0\\ \Leftrightarrow \color{red}{55.} \left(\frac{16}{55}x^2-\frac{8}{5}x+\frac{11}{5}\right)=0 \color{red}{.55} \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)