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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(4x=-\frac{8}{11}x^2-\frac{11}{2}\)
2. \((3x+2)(4x-4)-x(3x-33)=-12\)
3. \(x(2x+23)=18(x+1)\)
4. \(2x^2-(15x+96)=x(x-11)\)
5. \(-(8+3x)=-x^2-(-36-4x)\)
6. \(x(x+14)=3(x-10)\)
7. \(17x^2-(9x-9)=x(x+15)\)
8. \(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}=0\)
9. \(x(9x-10)=2(x-2)\)
10. \(x(4x+45)=9(x-9)\)
11. \(10x^2-(20x+24)=9x(x-2)\)
12. \(-(14-22x)=-x^2-(-85-20x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(4x=-\frac{8}{11}x^2-\frac{11}{2} \\ \Leftrightarrow \frac{8}{11}x^2+4x+\frac{11}{2}=0 \\ \Leftrightarrow \color{red}{22.} \left(\frac{8}{11}x^2+4x+\frac{11}{2}\right)=0 \color{red}{.22} \\ \Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
2. \((3x+2)(4x-4)-x(3x-33)=-12\\ \Leftrightarrow 12x^2-12x+8x-8 -3x^2+33x+12=0 \\ \Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.9.4 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\ & = \frac{-18}{18} & & = \frac{-8}{18} \\ & = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
3. \(x(2x+23)=18(x+1) \\ \Leftrightarrow 2x^2+23x=18x+18 \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \(2x^2-(15x+96)=x(x-11) \\ \Leftrightarrow 2x^2-15x-96=x^2-11x \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
5. \(-(8+3x)=-x^2-(-36-4x) \\ \Leftrightarrow -8-3x=-x^2+36+4x \\ \Leftrightarrow x^2-7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt225}{2.1} & & = \frac{-(-7)+\sqrt225}{2.1} \\ & = \frac{-8}{2} & & = \frac{22}{2} \\ & = -4 & & = 11 \\ \\ V &= \Big\{ -4 ; 11 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+14)=3(x-10) \\ \Leftrightarrow x^2+14x=3x-30 \\ \Leftrightarrow x^2+11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt1}{2.1} & & = \frac{-11+\sqrt1}{2.1} \\ & = \frac{-12}{2} & & = \frac{-10}{2} \\ & = -6 & & = -5 \\ \\ V &= \Big\{ -6 ; -5 \Big\} & &\end{align} \\ -----------------\)
7. \(17x^2-(9x-9)=x(x+15) \\ \Leftrightarrow 17x^2-9x+9=x^2+15x \\ \Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.16} & & \\ & = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.88 & &\\ & = 361-352 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt9}{2.1} & & = \frac{-19+\sqrt9}{2.1} \\ & = \frac{-22}{2} & & = \frac{-16}{2} \\ & = -11 & & = -8 \\ \\ V &= \Big\{ -11 ; -8 \Big\} & &\end{align} \\ -----------------\)
9. \(x(9x-10)=2(x-2) \\ \Leftrightarrow 9x^2-10x=2x-4 \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(x(4x+45)=9(x-9) \\ \Leftrightarrow 4x^2+45x=9x-81 \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(10x^2-(20x+24)=9x(x-2) \\ \Leftrightarrow 10x^2-20x-24=9x^2-18x \\ \Leftrightarrow x^2-2x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt100}{2.1} & & = \frac{-(-2)+\sqrt100}{2.1} \\ & = \frac{-8}{2} & & = \frac{12}{2} \\ & = -4 & & = 6 \\ \\ V &= \Big\{ -4 ; 6 \Big\} & &\end{align} \\ -----------------\)
12. \(-(14-22x)=-x^2-(-85-20x) \\ \Leftrightarrow -14+22x=-x^2+85+20x \\ \Leftrightarrow x^2+2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt400}{2.1} & & = \frac{-2+\sqrt400}{2.1} \\ & = \frac{-22}{2} & & = \frac{18}{2} \\ & = -11 & & = 9 \\ \\ V &= \Big\{ -11 ; 9 \Big\} & &\end{align} \\ -----------------\)