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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(4x+19)=4(x+1)\)
2. \(x(16x+7)=-(x+1)\)
3. \(x(x-8)=3(x-8)\)
4. \(x(18x+29)=4(x-2)\)
5. \((x+4)(x-1)-x(-15x+19)=-13\)
6. \(x(16x+27)=3(x-3)\)
7. \(5x^2-(4x-121)=x(x+20)\)
8. \(-\frac{1}{5}x=-\frac{1}{40}x^2-\frac{2}{5}\)
9. \(\frac{1}{5}x^2+\frac{1}{5}x+\frac{1}{20}=0\)
10. \(-(8+2x)=-4x^2-(17-10x)\)
11. \(-2x=-\frac{1}{2}x^2-\frac{81}{2}\)
12. \(\frac{2}{5}x=-\frac{1}{35}x^2-\frac{7}{5}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(4x+19)=4(x+1) \\ \Leftrightarrow 4x^2+19x=4x+4 \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(x(16x+7)=-(x+1) \\ \Leftrightarrow 16x^2+7x=-x-1 \\ \Leftrightarrow 16x^2+8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.16} & & \\ & = -\frac{1}{4} & & \\V &= \Big\{ -\frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(x(x-8)=3(x-8) \\ \Leftrightarrow x^2-8x=3x-24 \\ \Leftrightarrow x^2-11x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.24 & &\\ & = 121-96 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt25}{2.1} & & = \frac{-(-11)+\sqrt25}{2.1} \\ & = \frac{6}{2} & & = \frac{16}{2} \\ & = 3 & & = 8 \\ \\ V &= \Big\{ 3 ; 8 \Big\} & &\end{align} \\ -----------------\)
4. \(x(18x+29)=4(x-2) \\ \Leftrightarrow 18x^2+29x=4x-8 \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \((x+4)(x-1)-x(-15x+19)=-13\\ \Leftrightarrow x^2-x+4x-4 +15x^2-19x+13=0 \\ \Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.16} & & \\ & = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(x(16x+27)=3(x-3) \\ \Leftrightarrow 16x^2+27x=3x-9 \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(5x^2-(4x-121)=x(x+20) \\ \Leftrightarrow 5x^2-4x+121=x^2+20x \\ \Leftrightarrow 4x^2-24x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-24x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.4.121 & &\\ & = 576-1936 & & \\ & = -1360 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-\frac{1}{5}x=-\frac{1}{40}x^2-\frac{2}{5} \\ \Leftrightarrow \frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}=0 \\ \Leftrightarrow \color{red}{40.} \left(\frac{1}{40}x^2-\frac{1}{5}x+\frac{2}{5}\right)=0 \color{red}{.40} \\ \Leftrightarrow 9x^2-72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{5}x^2+\frac{1}{5}x+\frac{1}{20}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{1}{5}x+\frac{1}{20}\right)=0 \color{red}{.20} \\ \Leftrightarrow 16x^2+16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(-(8+2x)=-4x^2-(17-10x) \\ \Leftrightarrow -8-2x=-4x^2-17+10x \\ \Leftrightarrow 4x^2-12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.4} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(-2x=-\frac{1}{2}x^2-\frac{81}{2} \\ \Leftrightarrow \frac{1}{2}x^2-2x+\frac{81}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-2x+\frac{81}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-4x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.81 & &\\ & = 16-324 & & \\ & = -308 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(\frac{2}{5}x=-\frac{1}{35}x^2-\frac{7}{5} \\ \Leftrightarrow \frac{1}{35}x^2+\frac{2}{5}x+\frac{7}{5}=0 \\ \Leftrightarrow \color{red}{35.} \left(\frac{1}{35}x^2+\frac{2}{5}x+\frac{7}{5}\right)=0 \color{red}{.35} \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)