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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(2x+19)=2(x-4)\)
2. \(x(16x+16)=-(x+1)\)
3. \(2x^2-(9x+7)=x(x-3)\)
4. \(19x^2-(12x+2)=x(x-17)\)
5. \(\frac{5}{2}x=-2x^2+\frac{9}{2}\)
6. \(x(x-12)=10(x-12)\)
7. \((3x-3)(3x+3)-x(-7x-16)=-13\)
8. \(x=-\frac{1}{4}x^2-\frac{3}{4}\)
9. \(-(8-28x)=-x^2-(96-9x)\)
10. \(\frac{13}{8}x=-\frac{9}{4}x^2-\frac{1}{4}\)
11. \(x=-\frac{1}{11}x^2-\frac{11}{4}\)
12. \((-2x-2)(-4x+2)-x(7x-6)=116\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(2x+19)=2(x-4) \\ \Leftrightarrow 2x^2+19x=2x-8 \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(x(16x+16)=-(x+1) \\ \Leftrightarrow 16x^2+16x=-x-1 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
3. \(2x^2-(9x+7)=x(x-3) \\ \Leftrightarrow 2x^2-9x-7=x^2-3x \\ \Leftrightarrow x^2-6x-7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-7) & &\\ & = 36+28 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt64}{2.1} & & = \frac{-(-6)+\sqrt64}{2.1} \\ & = \frac{-2}{2} & & = \frac{14}{2} \\ & = -1 & & = 7 \\ \\ V &= \Big\{ -1 ; 7 \Big\} & &\end{align} \\ -----------------\)
4. \(19x^2-(12x+2)=x(x-17) \\ \Leftrightarrow 19x^2-12x-2=x^2-17x \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{5}{2}x=-2x^2+\frac{9}{2} \\ \Leftrightarrow 2x^2+\frac{5}{2}x-\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(2x^2+\frac{5}{2}x-\frac{9}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x-12)=10(x-12) \\ \Leftrightarrow x^2-12x=10x-120 \\ \Leftrightarrow x^2-22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-22)-\sqrt4}{2.1} & & = \frac{-(-22)+\sqrt4}{2.1} \\ & = \frac{20}{2} & & = \frac{24}{2} \\ & = 10 & & = 12 \\ \\ V &= \Big\{ 10 ; 12 \Big\} & &\end{align} \\ -----------------\)
7. \((3x-3)(3x+3)-x(-7x-16)=-13\\ \Leftrightarrow 9x^2+9x-9x-9 +7x^2+16x+13=0 \\ \Leftrightarrow 16x^2+16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.16} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x=-\frac{1}{4}x^2-\frac{3}{4} \\ \Leftrightarrow \frac{1}{4}x^2+x+\frac{3}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+x+\frac{3}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+4x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt4}{2.1} & & = \frac{-4+\sqrt4}{2.1} \\ & = \frac{-6}{2} & & = \frac{-2}{2} \\ & = -3 & & = -1 \\ \\ V &= \Big\{ -3 ; -1 \Big\} & &\end{align} \\ -----------------\)
9. \(-(8-28x)=-x^2-(96-9x) \\ \Leftrightarrow -8+28x=-x^2-96+9x \\ \Leftrightarrow x^2+19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.88 & &\\ & = 361-352 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt9}{2.1} & & = \frac{-19+\sqrt9}{2.1} \\ & = \frac{-22}{2} & & = \frac{-16}{2} \\ & = -11 & & = -8 \\ \\ V &= \Big\{ -11 ; -8 \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{13}{8}x=-\frac{9}{4}x^2-\frac{1}{4} \\ \Leftrightarrow \frac{9}{4}x^2+\frac{13}{8}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{4}x^2+\frac{13}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
11. \(x=-\frac{1}{11}x^2-\frac{11}{4} \\ \Leftrightarrow \frac{1}{11}x^2+x+\frac{11}{4}=0 \\ \Leftrightarrow \color{red}{44.} \left(\frac{1}{11}x^2+x+\frac{11}{4}\right)=0 \color{red}{.44} \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
12. \((-2x-2)(-4x+2)-x(7x-6)=116\\ \Leftrightarrow 8x^2-4x+8x-4 -7x^2+6x-116=0 \\ \Leftrightarrow x^2-2x-120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-120) & &\\ & = 4+480 & & \\ & = 484 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt484}{2.1} & & = \frac{-(-2)+\sqrt484}{2.1} \\ & = \frac{-20}{2} & & = \frac{24}{2} \\ & = -10 & & = 12 \\ \\ V &= \Big\{ -10 ; 12 \Big\} & &\end{align} \\ -----------------\)