VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer 

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(14x^2-(9x+2)=6x(x-4)\)
  2. \(10x^2-(15x-25)=x(x-45)\)
  3. \((x+1)(-5x+5)-x(-9x+5)=4\)
  4. \(\frac{1}{4}x^2+\frac{7}{24}x-1=0\)
  5. \(9x^2+\frac{7}{8}x-\frac{1}{4}=0\)
  6. \(x(x-7)=-3(x+1)\)
  7. \(10x^2-(20x-121)=x(x+44)\)
  8. \(\frac{1}{2}x^2-7x+\frac{45}{2}=0\)
  9. \((x+4)(-x-3)-x(-17x-71)=-61\)
  10. \(17x^2-(4x-49)=x(x-50)\)
  11. \(4x^2-(9x+54)=3x(x-2)\)
  12. \(-(9+7x)=-x^2-(33-7x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(14x^2-(9x+2)=6x(x-4) \\ \Leftrightarrow 14x^2-9x-2=6x^2-24x \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  2. \(10x^2-(15x-25)=x(x-45) \\ \Leftrightarrow 10x^2-15x+25=x^2-45x \\ \Leftrightarrow 9x^2+30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-30}{2.9} & & \\ & = -\frac{5}{3} & & \\V &= \Big\{ -\frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
  3. \((x+1)(-5x+5)-x(-9x+5)=4\\ \Leftrightarrow -5x^2+5x-5x+5 +9x^2-5x-4=0 \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{4}x^2+\frac{7}{24}x-1=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{4}x^2+\frac{7}{24}x-1\right)=0 \color{red}{.24} \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  5. \(9x^2+\frac{7}{8}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{8.} \left(9x^2+\frac{7}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  6. \(x(x-7)=-3(x+1) \\ \Leftrightarrow x^2-7x=-3x-3 \\ \Leftrightarrow x^2-4x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt4}{2.1} & & = \frac{-(-4)+\sqrt4}{2.1} \\ & = \frac{2}{2} & & = \frac{6}{2} \\ & = 1 & & = 3 \\ \\ V &= \Big\{ 1 ; 3 \Big\} & &\end{align} \\ -----------------\)
  7. \(10x^2-(20x-121)=x(x+44) \\ \Leftrightarrow 10x^2-20x+121=x^2+44x \\ \Leftrightarrow 9x^2-64x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-64x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-64)^2-4.9.121 & &\\ & = 4096-4356 & & \\ & = -260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \(\frac{1}{2}x^2-7x+\frac{45}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-7x+\frac{45}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-14x+45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.45 & &\\ & = 196-180 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt16}{2.1} & & = \frac{-(-14)+\sqrt16}{2.1} \\ & = \frac{10}{2} & & = \frac{18}{2} \\ & = 5 & & = 9 \\ \\ V &= \Big\{ 5 ; 9 \Big\} & &\end{align} \\ -----------------\)
  9. \((x+4)(-x-3)-x(-17x-71)=-61\\ \Leftrightarrow -x^2-3x-4x-12 +17x^2+71x+61=0 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(17x^2-(4x-49)=x(x-50) \\ \Leftrightarrow 17x^2-4x+49=x^2-50x \\ \Leftrightarrow 16x^2+46x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+46x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (46)^2-4.16.49 & &\\ & = 2116-3136 & & \\ & = -1020 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(4x^2-(9x+54)=3x(x-2) \\ \Leftrightarrow 4x^2-9x-54=3x^2-6x \\ \Leftrightarrow x^2-3x-54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt225}{2.1} & & = \frac{-(-3)+\sqrt225}{2.1} \\ & = \frac{-12}{2} & & = \frac{18}{2} \\ & = -6 & & = 9 \\ \\ V &= \Big\{ -6 ; 9 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(9+7x)=-x^2-(33-7x) \\ \Leftrightarrow -9-7x=-x^2-33+7x \\ \Leftrightarrow x^2-14x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.24 & &\\ & = 196-96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt100}{2.1} & & = \frac{-(-14)+\sqrt100}{2.1} \\ & = \frac{4}{2} & & = \frac{24}{2} \\ & = 2 & & = 12 \\ \\ V &= \Big\{ 2 ; 12 \Big\} & &\end{align} \\ -----------------\)
Oefeningengenerator wiskundeoefeningen.be 2026-03-24 21:06:59
Een site van Busleyden Atheneum Mechelen