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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-x-4)(4x-5)-x(-8x+20)=29\)
2. \(\frac{7}{2}x=-\frac{1}{2}x^2-3\)
3. \(\frac{1}{16}x^2-\frac{3}{4}x+\frac{5}{4}=0\)
4. \(\frac{1}{44}x^2-\frac{1}{2}x+\frac{11}{4}=0\)
5. \(-\frac{1}{2}x=-\frac{1}{4}x^2+6\)
6. \(-(6-10x)=-4x^2-(5-7x)\)
7. \(-(5+14x)=-x^2-(53-2x)\)
8. \(x(x+29)=9(x-11)\)
9. \(\frac{1}{3}x=-\frac{1}{60}x^2-\frac{5}{3}\)
10. \(x(4x+4)=-9(x+1)\)
11. \(19x^2-(16x-2)=x(x-29)\)
12. \((-5x-4)(-4x-5)-x(19x+61)=-44\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-x-4)(4x-5)-x(-8x+20)=29\\ \Leftrightarrow -4x^2+5x-16x+20 +8x^2-20x-29=0 \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{7}{2}x=-\frac{1}{2}x^2-3 \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{2}x+3=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{7}{2}x+3\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2+7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt25}{2.1} & & = \frac{-7+\sqrt25}{2.1} \\ & = \frac{-12}{2} & & = \frac{-2}{2} \\ & = -6 & & = -1 \\ \\ V &= \Big\{ -6 ; -1 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{16}x^2-\frac{3}{4}x+\frac{5}{4}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{3}{4}x+\frac{5}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2-12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.20 & &\\ & = 144-80 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt64}{2.1} & & = \frac{-(-12)+\sqrt64}{2.1} \\ & = \frac{4}{2} & & = \frac{20}{2} \\ & = 2 & & = 10 \\ \\ V &= \Big\{ 2 ; 10 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{44}x^2-\frac{1}{2}x+\frac{11}{4}=0\\ \Leftrightarrow \color{red}{44.} \left(\frac{1}{44}x^2-\frac{1}{2}x+\frac{11}{4}\right)=0 \color{red}{.44} \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
5. \(-\frac{1}{2}x=-\frac{1}{4}x^2+6 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{1}{2}x-6=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-6\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt100}{2.1} & & = \frac{-(-2)+\sqrt100}{2.1} \\ & = \frac{-8}{2} & & = \frac{12}{2} \\ & = -4 & & = 6 \\ \\ V &= \Big\{ -4 ; 6 \Big\} & &\end{align} \\ -----------------\)
6. \(-(6-10x)=-4x^2-(5-7x) \\ \Leftrightarrow -6+10x=-4x^2-5+7x \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(-(5+14x)=-x^2-(53-2x) \\ \Leftrightarrow -5-14x=-x^2-53+2x \\ \Leftrightarrow x^2-16x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.48 & &\\ & = 256-192 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt64}{2.1} & & = \frac{-(-16)+\sqrt64}{2.1} \\ & = \frac{8}{2} & & = \frac{24}{2} \\ & = 4 & & = 12 \\ \\ V &= \Big\{ 4 ; 12 \Big\} & &\end{align} \\ -----------------\)
8. \(x(x+29)=9(x-11) \\ \Leftrightarrow x^2+29x=9x-99 \\ \Leftrightarrow x^2+20x+99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.99 & &\\ & = 400-396 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-20-\sqrt4}{2.1} & & = \frac{-20+\sqrt4}{2.1} \\ & = \frac{-22}{2} & & = \frac{-18}{2} \\ & = -11 & & = -9 \\ \\ V &= \Big\{ -11 ; -9 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{3}x=-\frac{1}{60}x^2-\frac{5}{3} \\ \Leftrightarrow \frac{1}{60}x^2+\frac{1}{3}x+\frac{5}{3}=0 \\ \Leftrightarrow \color{red}{60.} \left(\frac{1}{60}x^2+\frac{1}{3}x+\frac{5}{3}\right)=0 \color{red}{.60} \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
10. \(x(4x+4)=-9(x+1) \\ \Leftrightarrow 4x^2+4x=-9x-9 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
11. \(19x^2-(16x-2)=x(x-29) \\ \Leftrightarrow 19x^2-16x+2=x^2-29x \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
12. \((-5x-4)(-4x-5)-x(19x+61)=-44\\ \Leftrightarrow 20x^2+25x+16x+20 -19x^2-61x+44=0 \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)