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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-5x-3)(4x+5)-x(-24x-8)=-79\)
2. \(-\frac{16}{3}x=-\frac{1}{3}x^2-\frac{121}{3}\)
3. \(-(4-16x)=-2x^2-(22-3x)\)
4. \(x(x+0)=-4(x+1)\)
5. \(\frac{1}{4}x^2-\frac{1}{4}x+\frac{1}{16}=0\)
6. \(x(18x+11)=-2(x+1)\)
7. \((-x+1)(x+2)-x(-5x+16)=-98\)
8. \(-(4-9x)=-72x^2-(2-2x)\)
9. \(x(16x+16)=-(x+1)\)
10. \(7x^2-(2x-24)=x(x-27)\)
11. \(-(8-35x)=-4x^2-(89-13x)\)
12. \(8x^2-(8x+48)=5x(x-3)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-5x-3)(4x+5)-x(-24x-8)=-79\\ \Leftrightarrow -20x^2-25x-12x-15 +24x^2+8x+79=0 \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
2. \(-\frac{16}{3}x=-\frac{1}{3}x^2-\frac{121}{3} \\ \Leftrightarrow \frac{1}{3}x^2-\frac{16}{3}x+\frac{121}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{16}{3}x+\frac{121}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-16x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.121 & &\\ & = 256-484 & & \\ & = -228 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(-(4-16x)=-2x^2-(22-3x) \\ \Leftrightarrow -4+16x=-2x^2-22+3x \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
4. \(x(x+0)=-4(x+1) \\ \Leftrightarrow x^2+0x=-4x-4 \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{4}x^2-\frac{1}{4}x+\frac{1}{16}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2-\frac{1}{4}x+\frac{1}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(x(18x+11)=-2(x+1) \\ \Leftrightarrow 18x^2+11x=-2x-2 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
7. \((-x+1)(x+2)-x(-5x+16)=-98\\ \Leftrightarrow -x^2-2x+x+2 +5x^2-16x+98=0 \\ \Leftrightarrow 4x^2-16x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.100 & &\\ & = 256-1600 & & \\ & = -1344 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-(4-9x)=-72x^2-(2-2x) \\ \Leftrightarrow -4+9x=-72x^2-2+2x \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(x(16x+16)=-(x+1) \\ \Leftrightarrow 16x^2+16x=-x-1 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
10. \(7x^2-(2x-24)=x(x-27) \\ \Leftrightarrow 7x^2-2x+24=x^2-27x \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(-(8-35x)=-4x^2-(89-13x) \\ \Leftrightarrow -8+35x=-4x^2-89+13x \\ \Leftrightarrow 4x^2+22x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+22x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.4.81 & &\\ & = 484-1296 & & \\ & = -812 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(8x^2-(8x+48)=5x(x-3) \\ \Leftrightarrow 8x^2-8x-48=5x^2-15x \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)