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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(9-26x)=-4x^2-(5-11x)\)
2. \(x(48x+10)=3(x+1)\)
3. \(\frac{1}{3}x=-\frac{1}{24}x^2+2\)
4. \(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}=0\)
5. \(-(12-18x)=-12x^2-(15-5x)\)
6. \(17x^2-(3x-81)=x(x+69)\)
7. \(x(2x+5)=2(x+1)\)
8. \(x(x-15)=-7(x+1)\)
9. \(x(x+25)=28(x+1)\)
10. \(x(x-12)=9(x-12)\)
11. \(\frac{9}{40}x^2+\frac{3}{5}x+\frac{2}{5}=0\)
12. \(3x^2-(4x+8)=x(x-19)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(9-26x)=-4x^2-(5-11x) \\ \Leftrightarrow -9+26x=-4x^2-5+11x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(x(48x+10)=3(x+1) \\ \Leftrightarrow 48x^2+10x=3x+3 \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{3}x=-\frac{1}{24}x^2+2 \\ \Leftrightarrow \frac{1}{24}x^2+\frac{1}{3}x-2=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{24}x^2+\frac{1}{3}x-2\right)=0 \color{red}{.24} \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2-\frac{1}{5}x-\frac{18}{5}\right)=0 \color{red}{.15} \\ \Leftrightarrow x^2-3x-54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt225}{2.1} & & = \frac{-(-3)+\sqrt225}{2.1} \\ & = \frac{-12}{2} & & = \frac{18}{2} \\ & = -6 & & = 9 \\ \\ V &= \Big\{ -6 ; 9 \Big\} & &\end{align} \\ -----------------\)
5. \(-(12-18x)=-12x^2-(15-5x) \\ \Leftrightarrow -12+18x=-12x^2-15+5x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(17x^2-(3x-81)=x(x+69) \\ \Leftrightarrow 17x^2-3x+81=x^2+69x \\ \Leftrightarrow 16x^2-72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.16} & & \\ & = \frac{9}{4} & & \\V &= \Big\{ \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(x(2x+5)=2(x+1) \\ \Leftrightarrow 2x^2+5x=2x+2 \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x(x-15)=-7(x+1) \\ \Leftrightarrow x^2-15x=-7x-7 \\ \Leftrightarrow x^2-8x+7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt36}{2.1} & & = \frac{-(-8)+\sqrt36}{2.1} \\ & = \frac{2}{2} & & = \frac{14}{2} \\ & = 1 & & = 7 \\ \\ V &= \Big\{ 1 ; 7 \Big\} & &\end{align} \\ -----------------\)
9. \(x(x+25)=28(x+1) \\ \Leftrightarrow x^2+25x=28x+28 \\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
10. \(x(x-12)=9(x-12) \\ \Leftrightarrow x^2-12x=9x-108 \\ \Leftrightarrow x^2-21x+108=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+108=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.108 & &\\ & = 441-432 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt9}{2.1} & & = \frac{-(-21)+\sqrt9}{2.1} \\ & = \frac{18}{2} & & = \frac{24}{2} \\ & = 9 & & = 12 \\ \\ V &= \Big\{ 9 ; 12 \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{9}{40}x^2+\frac{3}{5}x+\frac{2}{5}=0\\ \Leftrightarrow \color{red}{40.} \left(\frac{9}{40}x^2+\frac{3}{5}x+\frac{2}{5}\right)=0 \color{red}{.40} \\ \Leftrightarrow 9x^2+24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.9} & & \\ & = -\frac{4}{3} & & \\V &= \Big\{ -\frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
12. \(3x^2-(4x+8)=x(x-19) \\ \Leftrightarrow 3x^2-4x-8=x^2-19x \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)