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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{5}x^2+\frac{4}{5}x-\frac{21}{5}=0\)
2. \((-x-3)(3x+4)-x(-19x-112)=-156\)
3. \(-(3-10x)=-x^2-(-107-9x)\)
4. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\)
5. \(-(15-28x)=-4x^2-(51-10x)\)
6. \((2x-4)(-4x-2)-x(-12x-20)=-28\)
7. \(5x^2-(2x+1)=x(x-5)\)
8. \(x(x+0)=9(x-2)\)
9. \(-(9-57x)=-16x^2-(45-9x)\)
10. \(\frac{9}{32}x^2+\frac{3}{4}x+\frac{1}{2}=0\)
11. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4}\)
12. \(-(15-13x)=-x^2-(24-19x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{5}x^2+\frac{4}{5}x-\frac{21}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{4}{5}x-\frac{21}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+4x-21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt100}{2.1} & & = \frac{-4+\sqrt100}{2.1} \\ & = \frac{-14}{2} & & = \frac{6}{2} \\ & = -7 & & = 3 \\ \\ V &= \Big\{ -7 ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \((-x-3)(3x+4)-x(-19x-112)=-156\\ \Leftrightarrow -3x^2-4x-9x-12 +19x^2+112x+156=0 \\ \Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (96)^2-4.16.144 & &\\ & = 9216-9216 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-96}{2.16} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
3. \(-(3-10x)=-x^2-(-107-9x) \\ \Leftrightarrow -3+10x=-x^2+107+9x \\ \Leftrightarrow x^2+x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-110) & &\\ & = 1+440 & & \\ & = 441 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt441}{2.1} & & = \frac{-1+\sqrt441}{2.1} \\ & = \frac{-22}{2} & & = \frac{20}{2} \\ & = -11 & & = 10 \\ \\ V &= \Big\{ -11 ; 10 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2+x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-18}{2.1} & & \\ & = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
5. \(-(15-28x)=-4x^2-(51-10x) \\ \Leftrightarrow -15+28x=-4x^2-51+10x \\ \Leftrightarrow 4x^2+18x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+18x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.4.36 & &\\ & = 324-576 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \((2x-4)(-4x-2)-x(-12x-20)=-28\\ \Leftrightarrow -8x^2-4x+16x+8 +12x^2+20x+28=0 \\ \Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.4} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
7. \(5x^2-(2x+1)=x(x-5) \\ \Leftrightarrow 5x^2-2x-1=x^2-5x \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(x(x+0)=9(x-2) \\ \Leftrightarrow x^2+0x=9x-18 \\ \Leftrightarrow x^2-9x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.18 & &\\ & = 81-72 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt9}{2.1} & & = \frac{-(-9)+\sqrt9}{2.1} \\ & = \frac{6}{2} & & = \frac{12}{2} \\ & = 3 & & = 6 \\ \\ V &= \Big\{ 3 ; 6 \Big\} & &\end{align} \\ -----------------\)
9. \(-(9-57x)=-16x^2-(45-9x) \\ \Leftrightarrow -9+57x=-16x^2-45+9x \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{9}{32}x^2+\frac{3}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{9}{32}x^2+\frac{3}{4}x+\frac{1}{2}\right)=0 \color{red}{.32} \\ \Leftrightarrow 9x^2+24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.9} & & \\ & = -\frac{4}{3} & & \\V &= \Big\{ -\frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
12. \(-(15-13x)=-x^2-(24-19x) \\ \Leftrightarrow -15+13x=-x^2-24+19x \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)