VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(65x^2-(9x-3)=17x(x-2)\)
2. \(-\frac{13}{4}x=-\frac{1}{4}x^2-3\)
3. \(-\frac{4}{5}x=-\frac{1}{20}x^2-3\)
4. \(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}=0\)
5. \(-(5-8x)=-x^2-(9-10x)\)
6. \(x(x+33)=11(x-11)\)
7. \(-(13+70x)=-9x^2-(157-2x)\)
8. \(\frac{1}{3}x^2+\frac{7}{6}x-12=0\)
9. \(4x^2-(12x-9)=3x(x-6)\)
10. \(2x^2-(10x-36)=x(x-22)\)
11. \(-(12-36x)=-2x^2-(20-19x)\)
12. \(-(4-12x)=-18x^2-(2-7x)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(65x^2-(9x-3)=17x(x-2) \\ \Leftrightarrow 65x^2-9x+3=17x^2-34x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(-\frac{13}{4}x=-\frac{1}{4}x^2-3 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{13}{4}x+3=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{13}{4}x+3\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.12 & &\\ & = 169-48 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt121}{2.1} & & = \frac{-(-13)+\sqrt121}{2.1} \\ & = \frac{2}{2} & & = \frac{24}{2} \\ & = 1 & & = 12 \\ \\ V &= \Big\{ 1 ; 12 \Big\} & &\end{align} \\ -----------------\)
3. \(-\frac{4}{5}x=-\frac{1}{20}x^2-3 \\ \Leftrightarrow \frac{1}{20}x^2-\frac{4}{5}x+3=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{20}x^2-\frac{4}{5}x+3\right)=0 \color{red}{.20} \\ \Leftrightarrow x^2-16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt16}{2.1} & & = \frac{-(-16)+\sqrt16}{2.1} \\ & = \frac{12}{2} & & = \frac{20}{2} \\ & = 6 & & = 10 \\ \\ V &= \Big\{ 6 ; 10 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}=0\\ \Leftrightarrow \color{red}{40.} \left(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}\right)=0 \color{red}{.40} \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
5. \(-(5-8x)=-x^2-(9-10x) \\ \Leftrightarrow -5+8x=-x^2-9+10x \\ \Leftrightarrow x^2-2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.4 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(x(x+33)=11(x-11) \\ \Leftrightarrow x^2+33x=11x-121 \\ \Leftrightarrow x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-22}{2.1} & & \\ & = -11 & & \\V &= \Big\{ -11 \Big\} & &\end{align} \\ -----------------\)
7. \(-(13+70x)=-9x^2-(157-2x) \\ \Leftrightarrow -13-70x=-9x^2-157+2x \\ \Leftrightarrow 9x^2-72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{3}x^2+\frac{7}{6}x-12=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{7}{6}x-12\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(4x^2-(12x-9)=3x(x-6) \\ \Leftrightarrow 4x^2-12x+9=3x^2-18x \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
10. \(2x^2-(10x-36)=x(x-22) \\ \Leftrightarrow 2x^2-10x+36=x^2-22x \\ \Leftrightarrow x^2+12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.1} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
11. \(-(12-36x)=-2x^2-(20-19x) \\ \Leftrightarrow -12+36x=-2x^2-20+19x \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(-(4-12x)=-18x^2-(2-7x) \\ \Leftrightarrow -4+12x=-18x^2-2+7x \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)