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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(15-43x)=-18x^2-(23-18x)\)
2. \(-(2-38x)=-36x^2-(6-13x)\)
3. \(\frac{1}{4}x^2-\frac{11}{4}x+6=0\)
4. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2}\)
5. \((x+5)(-4x+1)-x(-5x+11)=41\)
6. \(5x^2-(4x+1)=x(x-7)\)
7. \((3x-3)(-3x-2)-x(-13x-5)=15\)
8. \(x^2-\frac{10}{3}x+\frac{49}{9}=0\)
9. \((2x+5)(-5x+5)-x(-11x+54)=-59\)
10. \(-\frac{2}{5}x=-\frac{1}{5}x^2-\frac{5}{4}\)
11. \(-(15-3x)=-9x^2-(51-5x)\)
12. \((3x-3)(-3x+3)-x(-25x+80)=-109\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(15-43x)=-18x^2-(23-18x) \\ \Leftrightarrow -15+43x=-18x^2-23+18x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(-(2-38x)=-36x^2-(6-13x) \\ \Leftrightarrow -2+38x=-36x^2-6+13x \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{4}x^2-\frac{11}{4}x+6=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{11}{4}x+6\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-11x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.24 & &\\ & = 121-96 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt25}{2.1} & & = \frac{-(-11)+\sqrt25}{2.1} \\ & = \frac{6}{2} & & = \frac{16}{2} \\ & = 3 & & = 8 \\ \\ V &= \Big\{ 3 ; 8 \Big\} & &\end{align} \\ -----------------\)
4. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-64)^2-4.16.64 & &\\ & = 4096-4096 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-64)}{2.16} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
5. \((x+5)(-4x+1)-x(-5x+11)=41\\ \Leftrightarrow -4x^2+x-20x+5 +5x^2-11x-41=0 \\ \Leftrightarrow x^2-5x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt169}{2.1} & & = \frac{-(-5)+\sqrt169}{2.1} \\ & = \frac{-8}{2} & & = \frac{18}{2} \\ & = -4 & & = 9 \\ \\ V &= \Big\{ -4 ; 9 \Big\} & &\end{align} \\ -----------------\)
6. \(5x^2-(4x+1)=x(x-7) \\ \Leftrightarrow 5x^2-4x-1=x^2-7x \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \((3x-3)(-3x-2)-x(-13x-5)=15\\ \Leftrightarrow -9x^2-6x+9x+6 +13x^2+5x-15=0 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
8. \(x^2-\frac{10}{3}x+\frac{49}{9}=0\\ \Leftrightarrow \color{red}{9.} \left(x^2-\frac{10}{3}x+\frac{49}{9}\right)=0 \color{red}{.9} \\ \Leftrightarrow 9x^2-30x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.49 & &\\ & = 900-1764 & & \\ & = -864 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \((2x+5)(-5x+5)-x(-11x+54)=-59\\ \Leftrightarrow -10x^2+10x-25x+25 +11x^2-54x+59=0 \\ \Leftrightarrow x^2-19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt25}{2.1} & & = \frac{-(-19)+\sqrt25}{2.1} \\ & = \frac{14}{2} & & = \frac{24}{2} \\ & = 7 & & = 12 \\ \\ V &= \Big\{ 7 ; 12 \Big\} & &\end{align} \\ -----------------\)
10. \(-\frac{2}{5}x=-\frac{1}{5}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{2}{5}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2-\frac{2}{5}x+\frac{5}{4}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2-8x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-8x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.4.25 & &\\ & = 64-400 & & \\ & = -336 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(-(15-3x)=-9x^2-(51-5x) \\ \Leftrightarrow -15+3x=-9x^2-51+5x \\ \Leftrightarrow 9x^2-2x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-2x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.9.36 & &\\ & = 4-1296 & & \\ & = -1292 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \((3x-3)(-3x+3)-x(-25x+80)=-109\\ \Leftrightarrow -9x^2+9x+9x-9 +25x^2-80x+109=0 \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)