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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{5}x^2+\frac{2}{5}x+\frac{81}{20}=0\)
2. \(-(6-18x)=-16x^2-(5-12x)\)
3. \(\frac{2}{7}x^2+2x+\frac{7}{2}=0\)
4. \(x(4x+15)=3(x-3)\)
5. \(\frac{5}{2}x=-\frac{9}{2}x^2+2\)
6. \(-\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2}\)
7. \(\frac{1}{80}x^2-\frac{1}{4}x+\frac{5}{4}=0\)
8. \(-\frac{11}{8}x=-\frac{1}{4}x^2-4\)
9. \(x(16x+12)=-4(x+1)\)
10. \(x(x-35)=-49(x+1)\)
11. \(-(2-38x)=-16x^2-(11-14x)\)
12. \((-5x-3)(-5x-3)-x(24x+40)=-46\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{5}x^2+\frac{2}{5}x+\frac{81}{20}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{2}{5}x+\frac{81}{20}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+8x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+8x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.4.81 & &\\ & = 64-1296 & & \\ & = -1232 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(-(6-18x)=-16x^2-(5-12x) \\ \Leftrightarrow -6+18x=-16x^2-5+12x \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{2}{7}x^2+2x+\frac{7}{2}=0\\ \Leftrightarrow \color{red}{14.} \left(\frac{2}{7}x^2+2x+\frac{7}{2}\right)=0 \color{red}{.14} \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(x(4x+15)=3(x-3) \\ \Leftrightarrow 4x^2+15x=3x-9 \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{5}{2}x=-\frac{9}{2}x^2+2 \\ \Leftrightarrow \frac{9}{2}x^2+\frac{5}{2}x-2=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{9}{2}x^2+\frac{5}{2}x-2\right)=0 \color{red}{.2} \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
6. \(-\frac{11}{4}x=-\frac{1}{4}x^2-\frac{5}{2} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{11}{4}x+\frac{5}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{11}{4}x+\frac{5}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\ & = \frac{2}{2} & & = \frac{20}{2} \\ & = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{80}x^2-\frac{1}{4}x+\frac{5}{4}=0\\ \Leftrightarrow \color{red}{80.} \left(\frac{1}{80}x^2-\frac{1}{4}x+\frac{5}{4}\right)=0 \color{red}{.80} \\ \Leftrightarrow x^2-20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.1} & & \\ & = 10 & & \\V &= \Big\{ 10 \Big\} & &\end{align} \\ -----------------\)
8. \(-\frac{11}{8}x=-\frac{1}{4}x^2-4 \\ \Leftrightarrow \frac{1}{4}x^2-\frac{11}{8}x+4=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2-\frac{11}{8}x+4\right)=0 \color{red}{.8} \\ \Leftrightarrow 4x^2-22x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-22x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.4.64 & &\\ & = 484-1024 & & \\ & = -540 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(x(16x+12)=-4(x+1) \\ \Leftrightarrow 16x^2+12x=-4x-4 \\ \Leftrightarrow 16x^2+16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(x(x-35)=-49(x+1) \\ \Leftrightarrow x^2-35x=-49x-49 \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
11. \(-(2-38x)=-16x^2-(11-14x) \\ \Leftrightarrow -2+38x=-16x^2-11+14x \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
12. \((-5x-3)(-5x-3)-x(24x+40)=-46\\ \Leftrightarrow 25x^2+15x+15x+9 -24x^2-40x+46=0 \\ \Leftrightarrow x^2-16x+55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.55 & &\\ & = 256-220 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt36}{2.1} & & = \frac{-(-16)+\sqrt36}{2.1} \\ & = \frac{10}{2} & & = \frac{22}{2} \\ & = 5 & & = 11 \\ \\ V &= \Big\{ 5 ; 11 \Big\} & &\end{align} \\ -----------------\)