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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(13-16x)=-x^2-(-23-11x)\)
2. \(\frac{1}{4}x^2-\frac{1}{2}x-\frac{3}{4}=0\)
3. \((2x+5)(-3x+1)-x(-7x+5)=13\)
4. \(\frac{1}{15}x^2+\frac{2}{3}x+\frac{5}{3}=0\)
5. \(-(6-92x)=-9x^2-(150-20x)\)
6. \(4x^2+\frac{17}{4}x+\frac{1}{4}=0\)
7. \(5x^2-(5x-1)=x(x-9)\)
8. \(\frac{1}{4}x^2+\frac{5}{2}x+36=0\)
9. \(\frac{1}{4}x^2+\frac{3}{4}x-22=0\)
10. \(x(6x+31)=24(x+1)\)
11. \(\frac{1}{3}x^2+\frac{19}{3}x+28=0\)
12. \(5x^2-(19x+4)=x(x-34)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(13-16x)=-x^2-(-23-11x) \\ \Leftrightarrow -13+16x=-x^2+23+11x \\ \Leftrightarrow x^2+5x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.1} & & = \frac{-5+\sqrt169}{2.1} \\ & = \frac{-18}{2} & & = \frac{8}{2} \\ & = -9 & & = 4 \\ \\ V &= \Big\{ -9 ; 4 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{4}x^2-\frac{1}{2}x-\frac{3}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-\frac{3}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)
3. \((2x+5)(-3x+1)-x(-7x+5)=13\\ \Leftrightarrow -6x^2+2x-15x+5 +7x^2-5x-13=0 \\ \Leftrightarrow x^2+2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt36}{2.1} & & = \frac{-2+\sqrt36}{2.1} \\ & = \frac{-8}{2} & & = \frac{4}{2} \\ & = -4 & & = 2 \\ \\ V &= \Big\{ -4 ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{1}{15}x^2+\frac{2}{3}x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2+\frac{2}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 4x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.4} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
5. \(-(6-92x)=-9x^2-(150-20x) \\ \Leftrightarrow -6+92x=-9x^2-150+20x \\ \Leftrightarrow 9x^2+72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.9} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
6. \(4x^2+\frac{17}{4}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(4x^2+\frac{17}{4}x+\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
7. \(5x^2-(5x-1)=x(x-9) \\ \Leftrightarrow 5x^2-5x+1=x^2-9x \\ \Leftrightarrow 4x^2+4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{4}x^2+\frac{5}{2}x+36=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{5}{2}x+36\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+10x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.144 & &\\ & = 100-576 & & \\ & = -476 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(\frac{1}{4}x^2+\frac{3}{4}x-22=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{3}{4}x-22\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+3x-88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-88) & &\\ & = 9+352 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt361}{2.1} & & = \frac{-3+\sqrt361}{2.1} \\ & = \frac{-22}{2} & & = \frac{16}{2} \\ & = -11 & & = 8 \\ \\ V &= \Big\{ -11 ; 8 \Big\} & &\end{align} \\ -----------------\)
10. \(x(6x+31)=24(x+1) \\ \Leftrightarrow 6x^2+31x=24x+24 \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{3}x^2+\frac{19}{3}x+28=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{19}{3}x+28\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt25}{2.1} & & = \frac{-19+\sqrt25}{2.1} \\ & = \frac{-24}{2} & & = \frac{-14}{2} \\ & = -12 & & = -7 \\ \\ V &= \Big\{ -12 ; -7 \Big\} & &\end{align} \\ -----------------\)
12. \(5x^2-(19x+4)=x(x-34) \\ \Leftrightarrow 5x^2-19x-4=x^2-34x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)