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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(6-60x)=-9x^2-(106-20x)\)
2. \(-\frac{1}{2}x=-\frac{1}{16}x^2+\frac{5}{4}\)
3. \(x(x+13)=3(x-8)\)
4. \(17x^2-(7x-49)=x(x+19)\)
5. \(x(16x+81)=9(x-9)\)
6. \(\frac{1}{10}x^2-\frac{1}{5}x+\frac{8}{5}=0\)
7. \(\frac{5}{8}x=-\frac{1}{4}x^2-\frac{1}{4}\)
8. \(-(12-27x)=-9x^2-(16-14x)\)
9. \(\frac{5}{4}x=-\frac{1}{4}x^2-\frac{3}{2}\)
10. \(2x=-\frac{2}{7}x^2-\frac{7}{2}\)
11. \(4x^2-(11x+56)=3x(x-4)\)
12. \(-(9-20x)=-x^2-(73-4x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(6-60x)=-9x^2-(106-20x) \\ \Leftrightarrow -6+60x=-9x^2-106+20x \\ \Leftrightarrow 9x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.9.100 & &\\ & = 1600-3600 & & \\ & = -2000 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(-\frac{1}{2}x=-\frac{1}{16}x^2+\frac{5}{4} \\ \Leftrightarrow \frac{1}{16}x^2-\frac{1}{2}x-\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{2}x-\frac{5}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow x^2-8x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-20) & &\\ & = 64+80 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt144}{2.1} & & = \frac{-(-8)+\sqrt144}{2.1} \\ & = \frac{-4}{2} & & = \frac{20}{2} \\ & = -2 & & = 10 \\ \\ V &= \Big\{ -2 ; 10 \Big\} & &\end{align} \\ -----------------\)
3. \(x(x+13)=3(x-8) \\ \Leftrightarrow x^2+13x=3x-24 \\ \Leftrightarrow x^2+10x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.24 & &\\ & = 100-96 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt4}{2.1} & & = \frac{-10+\sqrt4}{2.1} \\ & = \frac{-12}{2} & & = \frac{-8}{2} \\ & = -6 & & = -4 \\ \\ V &= \Big\{ -6 ; -4 \Big\} & &\end{align} \\ -----------------\)
4. \(17x^2-(7x-49)=x(x+19) \\ \Leftrightarrow 17x^2-7x+49=x^2+19x \\ \Leftrightarrow 16x^2-26x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-26x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-26)^2-4.16.49 & &\\ & = 676-3136 & & \\ & = -2460 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x(16x+81)=9(x-9) \\ \Leftrightarrow 16x^2+81x=9x-81 \\ \Leftrightarrow 16x^2+72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.16} & & \\ & = -\frac{9}{4} & & \\V &= \Big\{ -\frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{1}{10}x^2-\frac{1}{5}x+\frac{8}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2-\frac{1}{5}x+\frac{8}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 4x^2-8x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-8x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.4.64 & &\\ & = 64-1024 & & \\ & = -960 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(\frac{5}{8}x=-\frac{1}{4}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(-(12-27x)=-9x^2-(16-14x) \\ \Leftrightarrow -12+27x=-9x^2-16+14x \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{5}{4}x=-\frac{1}{4}x^2-\frac{3}{2} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{5}{4}x+\frac{3}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{5}{4}x+\frac{3}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt1}{2.1} & & = \frac{-5+\sqrt1}{2.1} \\ & = \frac{-6}{2} & & = \frac{-4}{2} \\ & = -3 & & = -2 \\ \\ V &= \Big\{ -3 ; -2 \Big\} & &\end{align} \\ -----------------\)
10. \(2x=-\frac{2}{7}x^2-\frac{7}{2} \\ \Leftrightarrow \frac{2}{7}x^2+2x+\frac{7}{2}=0 \\ \Leftrightarrow \color{red}{14.} \left(\frac{2}{7}x^2+2x+\frac{7}{2}\right)=0 \color{red}{.14} \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(4x^2-(11x+56)=3x(x-4) \\ \Leftrightarrow 4x^2-11x-56=3x^2-12x \\ \Leftrightarrow x^2+x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt225}{2.1} & & = \frac{-1+\sqrt225}{2.1} \\ & = \frac{-16}{2} & & = \frac{14}{2} \\ & = -8 & & = 7 \\ \\ V &= \Big\{ -8 ; 7 \Big\} & &\end{align} \\ -----------------\)
12. \(-(9-20x)=-x^2-(73-4x) \\ \Leftrightarrow -9+20x=-x^2-73+4x \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)