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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{9}{2}x^2+\frac{7}{4}x-2=0\)
2. \(-(9+66x)=-16x^2-(109-14x)\)
3. \(-(7-10x)=-x^2-(5-11x)\)
4. \(5x^2-(7x-9)=x(x-19)\)
5. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{15}{4}\)
6. \((x+2)(3x-1)-x(-x-8)=-3\)
7. \(-(7-9x)=-24x^2-(1-2x)\)
8. \(\frac{1}{6}x^2+x+\frac{3}{2}=0\)
9. \(-\frac{1}{5}x=-\frac{2}{5}x^2-\frac{1}{40}\)
10. \(-x=-\frac{3}{2}x^2-\frac{1}{6}\)
11. \(2x^2+\frac{25}{4}x+\frac{9}{2}=0\)
12. \(14x^2-(19x-36)=5x(x-11)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{9}{2}x^2+\frac{7}{4}x-2=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{9}{2}x^2+\frac{7}{4}x-2\right)=0 \color{red}{.4} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(-(9+66x)=-16x^2-(109-14x) \\ \Leftrightarrow -9-66x=-16x^2-109+14x \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(-(7-10x)=-x^2-(5-11x) \\ \Leftrightarrow -7+10x=-x^2-5+11x \\ \Leftrightarrow x^2-x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt9}{2.1} & & = \frac{-(-1)+\sqrt9}{2.1} \\ & = \frac{-2}{2} & & = \frac{4}{2} \\ & = -1 & & = 2 \\ \\ V &= \Big\{ -1 ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \(5x^2-(7x-9)=x(x-19) \\ \Leftrightarrow 5x^2-7x+9=x^2-19x \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(-\frac{1}{2}x=-\frac{1}{4}x^2+\frac{15}{4} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{2}x-\frac{15}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-\frac{15}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-15) & &\\ & = 4+60 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt64}{2.1} & & = \frac{-(-2)+\sqrt64}{2.1} \\ & = \frac{-6}{2} & & = \frac{10}{2} \\ & = -3 & & = 5 \\ \\ V &= \Big\{ -3 ; 5 \Big\} & &\end{align} \\ -----------------\)
6. \((x+2)(3x-1)-x(-x-8)=-3\\ \Leftrightarrow 3x^2-x+6x-2 +x^2+8x+3=0 \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(-(7-9x)=-24x^2-(1-2x) \\ \Leftrightarrow -7+9x=-24x^2-1+2x \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{6}x^2+x+\frac{3}{2}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+x+\frac{3}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
9. \(-\frac{1}{5}x=-\frac{2}{5}x^2-\frac{1}{40} \\ \Leftrightarrow \frac{2}{5}x^2-\frac{1}{5}x+\frac{1}{40}=0 \\ \Leftrightarrow \color{red}{40.} \left(\frac{2}{5}x^2-\frac{1}{5}x+\frac{1}{40}\right)=0 \color{red}{.40} \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(-x=-\frac{3}{2}x^2-\frac{1}{6} \\ \Leftrightarrow \frac{3}{2}x^2-x+\frac{1}{6}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{3}{2}x^2-x+\frac{1}{6}\right)=0 \color{red}{.6} \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(2x^2+\frac{25}{4}x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{25}{4}x+\frac{9}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(14x^2-(19x-36)=5x(x-11) \\ \Leftrightarrow 14x^2-19x+36=5x^2-55x \\ \Leftrightarrow 9x^2+36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)