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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x+6)=2(x-2)\)
2. \(-(9-11x)=-x^2-(25-3x)\)
3. \(5x^2-(18x-9)=4x(x-3)\)
4. \(\frac{7}{3}x=-\frac{2}{3}x^2-\frac{49}{24}\)
5. \(x(16x-35)=5(x-5)\)
6. \(9x^2-(11x-110)=8x(x-4)\)
7. \((-4x-1)(2x-3)-x(-20x+10)=6\)
8. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4}\)
9. \(\frac{6}{5}x=-\frac{9}{55}x^2-\frac{11}{5}\)
10. \(-(8-85x)=-9x^2-(152-13x)\)
11. \(10x^2-(8x-88)=9x(x-3)\)
12. \(5x^2-(17x+4)=x(x-32)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x+6)=2(x-2) \\ \Leftrightarrow x^2+6x=2x-4 \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
2. \(-(9-11x)=-x^2-(25-3x) \\ \Leftrightarrow -9+11x=-x^2-25+3x \\ \Leftrightarrow x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.1} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
3. \(5x^2-(18x-9)=4x(x-3) \\ \Leftrightarrow 5x^2-18x+9=4x^2-12x \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{7}{3}x=-\frac{2}{3}x^2-\frac{49}{24} \\ \Leftrightarrow \frac{2}{3}x^2+\frac{7}{3}x+\frac{49}{24}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2+\frac{7}{3}x+\frac{49}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
5. \(x(16x-35)=5(x-5) \\ \Leftrightarrow 16x^2-35x=5x-25 \\ \Leftrightarrow 16x^2-40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.16} & & \\ & = \frac{5}{4} & & \\V &= \Big\{ \frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(9x^2-(11x-110)=8x(x-4) \\ \Leftrightarrow 9x^2-11x+110=8x^2-32x \\ \Leftrightarrow x^2+21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-21-\sqrt1}{2.1} & & = \frac{-21+\sqrt1}{2.1} \\ & = \frac{-22}{2} & & = \frac{-20}{2} \\ & = -11 & & = -10 \\ \\ V &= \Big\{ -11 ; -10 \Big\} & &\end{align} \\ -----------------\)
7. \((-4x-1)(2x-3)-x(-20x+10)=6\\ \Leftrightarrow -8x^2+12x-2x+3 +20x^2-10x-6=0 \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{6}{5}x=-\frac{9}{55}x^2-\frac{11}{5} \\ \Leftrightarrow \frac{9}{55}x^2+\frac{6}{5}x+\frac{11}{5}=0 \\ \Leftrightarrow \color{red}{55.} \left(\frac{9}{55}x^2+\frac{6}{5}x+\frac{11}{5}\right)=0 \color{red}{.55} \\ \Leftrightarrow 9x^2+66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-66}{2.9} & & \\ & = -\frac{11}{3} & & \\V &= \Big\{ -\frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(-(8-85x)=-9x^2-(152-13x) \\ \Leftrightarrow -8+85x=-9x^2-152+13x \\ \Leftrightarrow 9x^2+72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.9} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
11. \(10x^2-(8x-88)=9x(x-3) \\ \Leftrightarrow 10x^2-8x+88=9x^2-27x \\ \Leftrightarrow x^2+19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.88 & &\\ & = 361-352 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt9}{2.1} & & = \frac{-19+\sqrt9}{2.1} \\ & = \frac{-22}{2} & & = \frac{-16}{2} \\ & = -11 & & = -8 \\ \\ V &= \Big\{ -11 ; -8 \Big\} & &\end{align} \\ -----------------\)
12. \(5x^2-(17x+4)=x(x-32) \\ \Leftrightarrow 5x^2-17x-4=x^2-32x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)