VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{3}{5}x=-\frac{1}{5}x^2-\frac{9}{20}\)
2. \(19x^2-(18x+8)=x(x-25)\)
3. \(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}=0\)
4. \(3x^2-(18x+77)=2x(x-7)\)
5. \((2x-5)(5x+3)-x(6x-13)=-79\)
6. \(4x^2-(6x+12)=x(x-11)\)
7. \(x(16x-68)=2(x-72)\)
8. \(-(7-26x)=-x^2-(32-18x)\)
9. \(x=-\frac{1}{5}x^2+\frac{6}{5}\)
10. \(2x^2-(10x-6)=x(x-17)\)
11. \(5x^2-(8x-144)=x(x+40)\)
12. \(\frac{4}{5}x^2+\frac{5}{6}x+\frac{1}{5}=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{3}{5}x=-\frac{1}{5}x^2-\frac{9}{20} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{3}{5}x+\frac{9}{20}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{3}{5}x+\frac{9}{20}\right)=0 \color{red}{.20} \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(19x^2-(18x+8)=x(x-25) \\ \Leftrightarrow 19x^2-18x-8=x^2-25x \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
4. \(3x^2-(18x+77)=2x(x-7) \\ \Leftrightarrow 3x^2-18x-77=2x^2-14x \\ \Leftrightarrow x^2-4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt324}{2.1} & & = \frac{-(-4)+\sqrt324}{2.1} \\ & = \frac{-14}{2} & & = \frac{22}{2} \\ & = -7 & & = 11 \\ \\ V &= \Big\{ -7 ; 11 \Big\} & &\end{align} \\ -----------------\)
5. \((2x-5)(5x+3)-x(6x-13)=-79\\ \Leftrightarrow 10x^2+6x-25x-15 -6x^2+13x+79=0 \\ \Leftrightarrow 4x^2+4x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.64 & &\\ & = 16-1024 & & \\ & = -1008 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(4x^2-(6x+12)=x(x-11) \\ \Leftrightarrow 4x^2-6x-12=x^2-11x \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(x(16x-68)=2(x-72) \\ \Leftrightarrow 16x^2-68x=2x-144 \\ \Leftrightarrow 16x^2-70x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-70x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-70)^2-4.16.144 & &\\ & = 4900-9216 & & \\ & = -4316 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-(7-26x)=-x^2-(32-18x) \\ \Leftrightarrow -7+26x=-x^2-32+18x \\ \Leftrightarrow x^2+8x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.25 & &\\ & = 64-100 & & \\ & = -36 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(x=-\frac{1}{5}x^2+\frac{6}{5} \\ \Leftrightarrow \frac{1}{5}x^2+x-\frac{6}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+x-\frac{6}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-6) & &\\ & = 25+24 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt49}{2.1} & & = \frac{-5+\sqrt49}{2.1} \\ & = \frac{-12}{2} & & = \frac{2}{2} \\ & = -6 & & = 1 \\ \\ V &= \Big\{ -6 ; 1 \Big\} & &\end{align} \\ -----------------\)
10. \(2x^2-(10x-6)=x(x-17) \\ \Leftrightarrow 2x^2-10x+6=x^2-17x \\ \Leftrightarrow x^2+7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt25}{2.1} & & = \frac{-7+\sqrt25}{2.1} \\ & = \frac{-12}{2} & & = \frac{-2}{2} \\ & = -6 & & = -1 \\ \\ V &= \Big\{ -6 ; -1 \Big\} & &\end{align} \\ -----------------\)
11. \(5x^2-(8x-144)=x(x+40) \\ \Leftrightarrow 5x^2-8x+144=x^2+40x \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{4}{5}x^2+\frac{5}{6}x+\frac{1}{5}=0\\ \Leftrightarrow \color{red}{30.} \left(\frac{4}{5}x^2+\frac{5}{6}x+\frac{1}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)