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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(11x^2-(20x+6)=5x(x-5)\)
2. \(\frac{1}{2}x^2-3x-\frac{7}{2}=0\)
3. \(3x^2-(9x+2)=2x(x-5)\)
4. \(\frac{5}{3}x=-\frac{4}{3}x^2+3\)
5. \((4x-5)(5x+2)-x(11x+4)=-11\)
6. \(-(3-28x)=-4x^2-(12-16x)\)
7. \((-x+5)(3x+4)-x(-4x+19)=60\)
8. \(\frac{1}{2}x^2+\frac{7}{4}x-18=0\)
9. \(-(6+x)=-x^2-(70-15x)\)
10. \(x(x+14)=2(x-10)\)
11. \(13x^2-(18x+18)=5x(x-5)\)
12. \(2x^2-(3x+120)=x(x-5)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(11x^2-(20x+6)=5x(x-5) \\ \Leftrightarrow 11x^2-20x-6=5x^2-25x \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{2}x^2-3x-\frac{7}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-3x-\frac{7}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt64}{2.1} & & = \frac{-(-6)+\sqrt64}{2.1} \\ & = \frac{-2}{2} & & = \frac{14}{2} \\ & = -1 & & = 7 \\ \\ V &= \Big\{ -1 ; 7 \Big\} & &\end{align} \\ -----------------\)
3. \(3x^2-(9x+2)=2x(x-5) \\ \Leftrightarrow 3x^2-9x-2=2x^2-10x \\ \Leftrightarrow x^2+x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-2) & &\\ & = 1+8 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt9}{2.1} & & = \frac{-1+\sqrt9}{2.1} \\ & = \frac{-4}{2} & & = \frac{2}{2} \\ & = -2 & & = 1 \\ \\ V &= \Big\{ -2 ; 1 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{5}{3}x=-\frac{4}{3}x^2+3 \\ \Leftrightarrow \frac{4}{3}x^2+\frac{5}{3}x-3=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+\frac{5}{3}x-3\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
5. \((4x-5)(5x+2)-x(11x+4)=-11\\ \Leftrightarrow 20x^2+8x-25x-10 -11x^2-4x+11=0 \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(-(3-28x)=-4x^2-(12-16x) \\ \Leftrightarrow -3+28x=-4x^2-12+16x \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
7. \((-x+5)(3x+4)-x(-4x+19)=60\\ \Leftrightarrow -3x^2-4x+15x+20 +4x^2-19x-60=0 \\ \Leftrightarrow x^2-3x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-40) & &\\ & = 9+160 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt169}{2.1} & & = \frac{-(-3)+\sqrt169}{2.1} \\ & = \frac{-10}{2} & & = \frac{16}{2} \\ & = -5 & & = 8 \\ \\ V &= \Big\{ -5 ; 8 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{2}x^2+\frac{7}{4}x-18=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{7}{4}x-18\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(-(6+x)=-x^2-(70-15x) \\ \Leftrightarrow -6-x=-x^2-70+15x \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
10. \(x(x+14)=2(x-10) \\ \Leftrightarrow x^2+14x=2x-20 \\ \Leftrightarrow x^2+12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.20 & &\\ & = 144-80 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt64}{2.1} & & = \frac{-12+\sqrt64}{2.1} \\ & = \frac{-20}{2} & & = \frac{-4}{2} \\ & = -10 & & = -2 \\ \\ V &= \Big\{ -10 ; -2 \Big\} & &\end{align} \\ -----------------\)
11. \(13x^2-(18x+18)=5x(x-5) \\ \Leftrightarrow 13x^2-18x-18=5x^2-25x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2-(3x+120)=x(x-5) \\ \Leftrightarrow 2x^2-3x-120=x^2-5x \\ \Leftrightarrow x^2+2x-120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-120) & &\\ & = 4+480 & & \\ & = 484 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt484}{2.1} & & = \frac{-2+\sqrt484}{2.1} \\ & = \frac{-24}{2} & & = \frac{20}{2} \\ & = -12 & & = 10 \\ \\ V &= \Big\{ -12 ; 10 \Big\} & &\end{align} \\ -----------------\)