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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x+8)=12(x+1)\)
2. \(24x^2-(7x+2)=6x(x-2)\)
3. \(2x^2-(17x-9)=x(x-23)\)
4. \(2x^2-(5x-25)=x(x-9)\)
5. \(x(4x-21)=-49(x+1)\)
6. \(-(15-43x)=-36x^2-(19-18x)\)
7. \(x(8x+15)=-2(x+1)\)
8. \(-\frac{3}{4}x=-\frac{9}{80}x^2-\frac{5}{4}\)
9. \(\frac{1}{5}x^2+\frac{1}{6}x-\frac{1}{5}=0\)
10. \(22x^2-(17x-64)=13x(x-5)\)
11. \(\frac{1}{32}x^2+\frac{1}{4}x-\frac{3}{2}=0\)
12. \(-(5-20x)=-8x^2-(-13-13x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x+8)=12(x+1) \\ \Leftrightarrow x^2+8x=12x+12 \\ \Leftrightarrow x^2-4x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\ & = \frac{-4}{2} & & = \frac{12}{2} \\ & = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
2. \(24x^2-(7x+2)=6x(x-2) \\ \Leftrightarrow 24x^2-7x-2=6x^2-12x \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
3. \(2x^2-(17x-9)=x(x-23) \\ \Leftrightarrow 2x^2-17x+9=x^2-23x \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
4. \(2x^2-(5x-25)=x(x-9) \\ \Leftrightarrow 2x^2-5x+25=x^2-9x \\ \Leftrightarrow x^2+4x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.25 & &\\ & = 16-100 & & \\ & = -84 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x(4x-21)=-49(x+1) \\ \Leftrightarrow 4x^2-21x=-49x-49 \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(-(15-43x)=-36x^2-(19-18x) \\ \Leftrightarrow -15+43x=-36x^2-19+18x \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(x(8x+15)=-2(x+1) \\ \Leftrightarrow 8x^2+15x=-2x-2 \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(-\frac{3}{4}x=-\frac{9}{80}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{9}{80}x^2-\frac{3}{4}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{80.} \left(\frac{9}{80}x^2-\frac{3}{4}x+\frac{5}{4}\right)=0 \color{red}{.80} \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{5}x^2+\frac{1}{6}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{30.} \left(\frac{1}{5}x^2+\frac{1}{6}x-\frac{1}{5}\right)=0 \color{red}{.30} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(22x^2-(17x-64)=13x(x-5) \\ \Leftrightarrow 22x^2-17x+64=13x^2-65x \\ \Leftrightarrow 9x^2+48x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+48x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.9.64 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.9} & & \\ & = -\frac{8}{3} & & \\V &= \Big\{ -\frac{8}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{32}x^2+\frac{1}{4}x-\frac{3}{2}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{32}x^2+\frac{1}{4}x-\frac{3}{2}\right)=0 \color{red}{.32} \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
12. \(-(5-20x)=-8x^2-(-13-13x) \\ \Leftrightarrow -5+20x=-8x^2+13+13x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)