VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(9-18x)=-x^2-(37-7x)\)
2. \(\frac{5}{3}x=-\frac{1}{5}x^2-\frac{16}{5}\)
3. \(-(12-3x)=-x^2-(-60-9x)\)
4. \(\frac{1}{15}x^2-\frac{2}{5}x+\frac{3}{5}=0\)
5. \(4x^2+\frac{3}{2}x-\frac{1}{4}=0\)
6. \(\frac{15}{8}x=-\frac{1}{2}x^2+\frac{1}{2}\)
7. \(3x^2-(6x+72)=x(x-13)\)
8. \(x(x+49)=55(x+1)\)
9. \(2x^2-(17x-120)=x(x+5)\)
10. \(\frac{1}{3}x^2+\frac{5}{6}x+\frac{1}{3}=0\)
11. \((2x-4)(-5x-2)-x(-11x+2)=16\)
12. \(x(x+11)=3(x-5)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(9-18x)=-x^2-(37-7x) \\ \Leftrightarrow -9+18x=-x^2-37+7x \\ \Leftrightarrow x^2+11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt9}{2.1} & & = \frac{-11+\sqrt9}{2.1} \\ & = \frac{-14}{2} & & = \frac{-8}{2} \\ & = -7 & & = -4 \\ \\ V &= \Big\{ -7 ; -4 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{5}{3}x=-\frac{1}{5}x^2-\frac{16}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{5}{3}x+\frac{16}{5}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{5}x^2+\frac{5}{3}x+\frac{16}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
3. \(-(12-3x)=-x^2-(-60-9x) \\ \Leftrightarrow -12+3x=-x^2+60+9x \\ \Leftrightarrow x^2-6x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-72) & &\\ & = 36+288 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt324}{2.1} & & = \frac{-(-6)+\sqrt324}{2.1} \\ & = \frac{-12}{2} & & = \frac{24}{2} \\ & = -6 & & = 12 \\ \\ V &= \Big\{ -6 ; 12 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{15}x^2-\frac{2}{5}x+\frac{3}{5}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2-\frac{2}{5}x+\frac{3}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow 9x^2-54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
5. \(4x^2+\frac{3}{2}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(4x^2+\frac{3}{2}x-\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{15}{8}x=-\frac{1}{2}x^2+\frac{1}{2} \\ \Leftrightarrow \frac{1}{2}x^2+\frac{15}{8}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{2}x^2+\frac{15}{8}x-\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(3x^2-(6x+72)=x(x-13) \\ \Leftrightarrow 3x^2-6x-72=x^2-13x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x(x+49)=55(x+1) \\ \Leftrightarrow x^2+49x=55x+55 \\ \Leftrightarrow x^2-6x-55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-55) & &\\ & = 36+220 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt256}{2.1} & & = \frac{-(-6)+\sqrt256}{2.1} \\ & = \frac{-10}{2} & & = \frac{22}{2} \\ & = -5 & & = 11 \\ \\ V &= \Big\{ -5 ; 11 \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2-(17x-120)=x(x+5) \\ \Leftrightarrow 2x^2-17x+120=x^2+5x \\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{1}{3}x^2+\frac{5}{6}x+\frac{1}{3}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{5}{6}x+\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
11. \((2x-4)(-5x-2)-x(-11x+2)=16\\ \Leftrightarrow -10x^2-4x+20x+8 +11x^2-2x-16=0 \\ \Leftrightarrow x^2+2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt36}{2.1} & & = \frac{-2+\sqrt36}{2.1} \\ & = \frac{-8}{2} & & = \frac{4}{2} \\ & = -4 & & = 2 \\ \\ V &= \Big\{ -4 ; 2 \Big\} & &\end{align} \\ -----------------\)
12. \(x(x+11)=3(x-5) \\ \Leftrightarrow x^2+11x=3x-15 \\ \Leftrightarrow x^2+8x+15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt4}{2.1} & & = \frac{-8+\sqrt4}{2.1} \\ & = \frac{-10}{2} & & = \frac{-6}{2} \\ & = -5 & & = -3 \\ \\ V &= \Big\{ -5 ; -3 \Big\} & &\end{align} \\ -----------------\)