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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(4x^2-(2x-6)=3x(x-3)\)
2. \(-(15-23x)=-9x^2-(96-7x)\)
3. \(-2x=-\frac{2}{11}x^2-\frac{11}{2}\)
4. \(-x=-\frac{1}{7}x^2-\frac{7}{4}\)
5. \(x(72x+9)=2(x+1)\)
6. \(6x^2-(2x-121)=2x(x-6)\)
7. \(x(12x+19)=12(x+1)\)
8. \(2x^2-(5x-144)=x(x+19)\)
9. \(x(16x+52)=4(x-9)\)
10. \(x(x+1)=4(x+1)\)
11. \(11x^2-(6x-1)=2x(x-2)\)
12. \(x(16x+45)=5(x-5)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(4x^2-(2x-6)=3x(x-3) \\ \Leftrightarrow 4x^2-2x+6=3x^2-9x \\ \Leftrightarrow x^2+7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt25}{2.1} & & = \frac{-7+\sqrt25}{2.1} \\ & = \frac{-12}{2} & & = \frac{-2}{2} \\ & = -6 & & = -1 \\ \\ V &= \Big\{ -6 ; -1 \Big\} & &\end{align} \\ -----------------\)
2. \(-(15-23x)=-9x^2-(96-7x) \\ \Leftrightarrow -15+23x=-9x^2-96+7x \\ \Leftrightarrow 9x^2+16x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+16x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.9.81 & &\\ & = 256-2916 & & \\ & = -2660 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(-2x=-\frac{2}{11}x^2-\frac{11}{2} \\ \Leftrightarrow \frac{2}{11}x^2-2x+\frac{11}{2}=0 \\ \Leftrightarrow \color{red}{22.} \left(\frac{2}{11}x^2-2x+\frac{11}{2}\right)=0 \color{red}{.22} \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(-x=-\frac{1}{7}x^2-\frac{7}{4} \\ \Leftrightarrow \frac{1}{7}x^2-x+\frac{7}{4}=0 \\ \Leftrightarrow \color{red}{28.} \left(\frac{1}{7}x^2-x+\frac{7}{4}\right)=0 \color{red}{.28} \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(x(72x+9)=2(x+1) \\ \Leftrightarrow 72x^2+9x=2x+2 \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(6x^2-(2x-121)=2x(x-6) \\ \Leftrightarrow 6x^2-2x+121=2x^2-12x \\ \Leftrightarrow 4x^2+10x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+10x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.4.121 & &\\ & = 100-1936 & & \\ & = -1836 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(x(12x+19)=12(x+1) \\ \Leftrightarrow 12x^2+19x=12x+12 \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(2x^2-(5x-144)=x(x+19) \\ \Leftrightarrow 2x^2-5x+144=x^2+19x \\ \Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.1} & & \\ & = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
9. \(x(16x+52)=4(x-9) \\ \Leftrightarrow 16x^2+52x=4x-36 \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(x(x+1)=4(x+1) \\ \Leftrightarrow x^2+x=4x+4 \\ \Leftrightarrow x^2-3x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-4) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt25}{2.1} & & = \frac{-(-3)+\sqrt25}{2.1} \\ & = \frac{-2}{2} & & = \frac{8}{2} \\ & = -1 & & = 4 \\ \\ V &= \Big\{ -1 ; 4 \Big\} & &\end{align} \\ -----------------\)
11. \(11x^2-(6x-1)=2x(x-2) \\ \Leftrightarrow 11x^2-6x+1=2x^2-4x \\ \Leftrightarrow 9x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.9.1 & &\\ & = 4-36 & & \\ & = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(x(16x+45)=5(x-5) \\ \Leftrightarrow 16x^2+45x=5x-25 \\ \Leftrightarrow 16x^2+40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.16} & & \\ & = -\frac{5}{4} & & \\V &= \Big\{ -\frac{5}{4} \Big\} & &\end{align} \\ -----------------\)