VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(12-10x)=-x^2-(61-2x)\)
2. \(\frac{13}{6}x=-\frac{1}{3}x^2-3\)
3. \(\frac{7}{6}x=-12x^2+\frac{1}{3}\)
4. \(x(16x-105)=-49(x+1)\)
5. \(\frac{7}{2}x=-\frac{1}{2}x^2+4\)
6. \((4x+3)(x-4)-x(-12x-20)=-13\)
7. \(2x^2-(20x-25)=x(x-10)\)
8. \(x(16x+63)=7(x-7)\)
9. \(2x^2-(19x-96)=x(x-39)\)
10. \((2x-3)(-2x+5)-x(-13x-10)=-11\)
11. \(x(12x+8)=3(x+1)\)
12. \(7x^2-(4x-6)=x(x-17)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(12-10x)=-x^2-(61-2x) \\ \Leftrightarrow -12+10x=-x^2-61+2x \\ \Leftrightarrow x^2+8x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.49 & &\\ & = 64-196 & & \\ & = -132 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(\frac{13}{6}x=-\frac{1}{3}x^2-3 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{13}{6}x+3=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{13}{6}x+3\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{7}{6}x=-12x^2+\frac{1}{3} \\ \Leftrightarrow 12x^2+\frac{7}{6}x-\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(12x^2+\frac{7}{6}x-\frac{1}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
4. \(x(16x-105)=-49(x+1) \\ \Leftrightarrow 16x^2-105x=-49x-49 \\ \Leftrightarrow 16x^2-56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-56)}{2.16} & & \\ & = \frac{7}{4} & & \\V &= \Big\{ \frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{7}{2}x=-\frac{1}{2}x^2+4 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{2}x-4=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{7}{2}x-4\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt81}{2.1} & & = \frac{-7+\sqrt81}{2.1} \\ & = \frac{-16}{2} & & = \frac{2}{2} \\ & = -8 & & = 1 \\ \\ V &= \Big\{ -8 ; 1 \Big\} & &\end{align} \\ -----------------\)
6. \((4x+3)(x-4)-x(-12x-20)=-13\\ \Leftrightarrow 4x^2-16x+3x-12 +12x^2+20x+13=0 \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(2x^2-(20x-25)=x(x-10) \\ \Leftrightarrow 2x^2-20x+25=x^2-10x \\ \Leftrightarrow x^2-10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
8. \(x(16x+63)=7(x-7) \\ \Leftrightarrow 16x^2+63x=7x-49 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2-(19x-96)=x(x-39) \\ \Leftrightarrow 2x^2-19x+96=x^2-39x \\ \Leftrightarrow x^2+20x+96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.96 & &\\ & = 400-384 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-20-\sqrt16}{2.1} & & = \frac{-20+\sqrt16}{2.1} \\ & = \frac{-24}{2} & & = \frac{-16}{2} \\ & = -12 & & = -8 \\ \\ V &= \Big\{ -12 ; -8 \Big\} & &\end{align} \\ -----------------\)
10. \((2x-3)(-2x+5)-x(-13x-10)=-11\\ \Leftrightarrow -4x^2+10x+6x-15 +13x^2+10x+11=0 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
11. \(x(12x+8)=3(x+1) \\ \Leftrightarrow 12x^2+8x=3x+3 \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
12. \(7x^2-(4x-6)=x(x-17) \\ \Leftrightarrow 7x^2-4x+6=x^2-17x \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)