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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{25}{12}x=-4x^2-\frac{1}{4}\)
2. \(x(4x+50)=10(x-10)\)
3. \((5x-4)(-2x-3)-x(-14x-6)=13\)
4. \(2x^2-(13x-16)=x(x-5)\)
5. \(x(x+32)=8(x-18)\)
6. \(10x^2-(15x-36)=x(x+21)\)
7. \(2x^2+\frac{17}{4}x+\frac{1}{2}=0\)
8. \(4x^2-(13x-24)=3x(x-9)\)
9. \((-2x-4)(-3x+2)-x(-10x-29)=-9\)
10. \(x(x+21)=3(x-27)\)
11. \(\frac{1}{2}x=-\frac{1}{5}x^2+\frac{9}{5}\)
12. \(-(6-28x)=-18x^2-(14-3x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{25}{12}x=-4x^2-\frac{1}{4} \\ \Leftrightarrow 4x^2+\frac{25}{12}x+\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{12.} \left(4x^2+\frac{25}{12}x+\frac{1}{4}\right)=0 \color{red}{.12} \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(x(4x+50)=10(x-10) \\ \Leftrightarrow 4x^2+50x=10x-100 \\ \Leftrightarrow 4x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.4} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
3. \((5x-4)(-2x-3)-x(-14x-6)=13\\ \Leftrightarrow -10x^2-15x+8x+12 +14x^2+6x-13=0 \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(2x^2-(13x-16)=x(x-5) \\ \Leftrightarrow 2x^2-13x+16=x^2-5x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
5. \(x(x+32)=8(x-18) \\ \Leftrightarrow x^2+32x=8x-144 \\ \Leftrightarrow x^2+24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.1} & & \\ & = -12 & & \\V &= \Big\{ -12 \Big\} & &\end{align} \\ -----------------\)
6. \(10x^2-(15x-36)=x(x+21) \\ \Leftrightarrow 10x^2-15x+36=x^2+21x \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
7. \(2x^2+\frac{17}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{17}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(4x^2-(13x-24)=3x(x-9) \\ \Leftrightarrow 4x^2-13x+24=3x^2-27x \\ \Leftrightarrow x^2+14x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.24 & &\\ & = 196-96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt100}{2.1} & & = \frac{-14+\sqrt100}{2.1} \\ & = \frac{-24}{2} & & = \frac{-4}{2} \\ & = -12 & & = -2 \\ \\ V &= \Big\{ -12 ; -2 \Big\} & &\end{align} \\ -----------------\)
9. \((-2x-4)(-3x+2)-x(-10x-29)=-9\\ \Leftrightarrow 6x^2-4x+12x-8 +10x^2+29x+9=0 \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
10. \(x(x+21)=3(x-27) \\ \Leftrightarrow x^2+21x=3x-81 \\ \Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-18}{2.1} & & \\ & = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{2}x=-\frac{1}{5}x^2+\frac{9}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{1}{2}x-\frac{9}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{1}{2}x-\frac{9}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
12. \(-(6-28x)=-18x^2-(14-3x) \\ \Leftrightarrow -6+28x=-18x^2-14+3x \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)