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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-4x+2)(5x+1)-x(-21x-8)=18\)
2. \(3x^2-(19x+28)=2x(x-11)\)
3. \((-5x-5)(5x+1)-x(-29x-13)=-4\)
4. \(\frac{1}{9}x^2+x+\frac{9}{4}=0\)
5. \(x(x+25)=21(x+1)\)
6. \(-\frac{3}{4}x=-\frac{1}{4}x^2-\frac{1}{2}\)
7. \(\frac{1}{6}x^2-\frac{5}{3}x-4=0\)
8. \(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}=0\)
9. \(\frac{1}{35}x^2+\frac{2}{5}x+\frac{7}{5}=0\)
10. \((-2x+3)(4x+2)-x(-9x+0)=30\)
11. \(x(36x+21)=-4(x+1)\)
12. \((-5x+4)(5x+2)-x(-26x+2)=4\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-4x+2)(5x+1)-x(-21x-8)=18\\ \Leftrightarrow -20x^2-4x+10x+2 +21x^2+8x-18=0 \\ \Leftrightarrow x^2+6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-16) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.1} & & = \frac{-6+\sqrt100}{2.1} \\ & = \frac{-16}{2} & & = \frac{4}{2} \\ & = -8 & & = 2 \\ \\ V &= \Big\{ -8 ; 2 \Big\} & &\end{align} \\ -----------------\)
2. \(3x^2-(19x+28)=2x(x-11) \\ \Leftrightarrow 3x^2-19x-28=2x^2-22x \\ \Leftrightarrow x^2+3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt121}{2.1} & & = \frac{-3+\sqrt121}{2.1} \\ & = \frac{-14}{2} & & = \frac{8}{2} \\ & = -7 & & = 4 \\ \\ V &= \Big\{ -7 ; 4 \Big\} & &\end{align} \\ -----------------\)
3. \((-5x-5)(5x+1)-x(-29x-13)=-4\\ \Leftrightarrow -25x^2-5x-25x-5 +29x^2+13x+4=0 \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{9}x^2+x+\frac{9}{4}=0\\ \Leftrightarrow \color{red}{36.} \left(\frac{1}{9}x^2+x+\frac{9}{4}\right)=0 \color{red}{.36} \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(x(x+25)=21(x+1) \\ \Leftrightarrow x^2+25x=21x+21 \\ \Leftrightarrow x^2+4x-21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt100}{2.1} & & = \frac{-4+\sqrt100}{2.1} \\ & = \frac{-14}{2} & & = \frac{6}{2} \\ & = -7 & & = 3 \\ \\ V &= \Big\{ -7 ; 3 \Big\} & &\end{align} \\ -----------------\)
6. \(-\frac{3}{4}x=-\frac{1}{4}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{3}{4}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{3}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-3x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.2 & &\\ & = 9-8 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt1}{2.1} & & = \frac{-(-3)+\sqrt1}{2.1} \\ & = \frac{2}{2} & & = \frac{4}{2} \\ & = 1 & & = 2 \\ \\ V &= \Big\{ 1 ; 2 \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{6}x^2-\frac{5}{3}x-4=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2-\frac{5}{3}x-4\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2-10x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt196}{2.1} & & = \frac{-(-10)+\sqrt196}{2.1} \\ & = \frac{-4}{2} & & = \frac{24}{2} \\ & = -2 & & = 12 \\ \\ V &= \Big\{ -2 ; 12 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{4}{5}x^2+\frac{3}{2}x-\frac{1}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{35}x^2+\frac{2}{5}x+\frac{7}{5}=0\\ \Leftrightarrow \color{red}{35.} \left(\frac{1}{35}x^2+\frac{2}{5}x+\frac{7}{5}\right)=0 \color{red}{.35} \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
10. \((-2x+3)(4x+2)-x(-9x+0)=30\\ \Leftrightarrow -8x^2-4x+12x+6 +9x^2+0x-30=0 \\ \Leftrightarrow x^2+2x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt100}{2.1} & & = \frac{-2+\sqrt100}{2.1} \\ & = \frac{-12}{2} & & = \frac{8}{2} \\ & = -6 & & = 4 \\ \\ V &= \Big\{ -6 ; 4 \Big\} & &\end{align} \\ -----------------\)
11. \(x(36x+21)=-4(x+1) \\ \Leftrightarrow 36x^2+21x=-4x-4 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
12. \((-5x+4)(5x+2)-x(-26x+2)=4\\ \Leftrightarrow -25x^2-10x+20x+8 +26x^2-2x-4=0 \\ \Leftrightarrow x^2-4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.1} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)