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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(44x^2-(7x-4)=8x(x-4)\)
2. \((-4x-5)(x-3)-x(-5x+31)=92\)
3. \(x(x-7)=5(x-7)\)
4. \(-(12-34x)=-4x^2-(93-2x)\)
5. \(5x^2-(10x+4)=x(x-25)\)
6. \(\frac{1}{6}x^2-x+\frac{3}{2}=0\)
7. \((4x-5)(-3x+1)-x(-16x+43)=-126\)
8. \((-2x-4)(3x-2)-x(-10x+9)=9\)
9. \(x=-\frac{1}{18}x^2-\frac{9}{2}\)
10. \(\frac{5}{8}x=-\frac{1}{4}x^2+\frac{9}{4}\)
11. \(x(x-6)=2(x-6)\)
12. \(x(9x-72)=-36(x+1)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(44x^2-(7x-4)=8x(x-4) \\ \Leftrightarrow 44x^2-7x+4=8x^2-32x \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \((-4x-5)(x-3)-x(-5x+31)=92\\ \Leftrightarrow -4x^2+12x-5x+15 +5x^2-31x-92=0 \\ \Leftrightarrow x^2-4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt324}{2.1} & & = \frac{-(-4)+\sqrt324}{2.1} \\ & = \frac{-14}{2} & & = \frac{22}{2} \\ & = -7 & & = 11 \\ \\ V &= \Big\{ -7 ; 11 \Big\} & &\end{align} \\ -----------------\)
3. \(x(x-7)=5(x-7) \\ \Leftrightarrow x^2-7x=5x-35 \\ \Leftrightarrow x^2-12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt4}{2.1} & & = \frac{-(-12)+\sqrt4}{2.1} \\ & = \frac{10}{2} & & = \frac{14}{2} \\ & = 5 & & = 7 \\ \\ V &= \Big\{ 5 ; 7 \Big\} & &\end{align} \\ -----------------\)
4. \(-(12-34x)=-4x^2-(93-2x) \\ \Leftrightarrow -12+34x=-4x^2-93+2x \\ \Leftrightarrow 4x^2+32x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+32x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.4.81 & &\\ & = 1024-1296 & & \\ & = -272 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(5x^2-(10x+4)=x(x-25) \\ \Leftrightarrow 5x^2-10x-4=x^2-25x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{1}{6}x^2-x+\frac{3}{2}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2-x+\frac{3}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
7. \((4x-5)(-3x+1)-x(-16x+43)=-126\\ \Leftrightarrow -12x^2+4x+15x-5 +16x^2-43x+126=0 \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
8. \((-2x-4)(3x-2)-x(-10x+9)=9\\ \Leftrightarrow -6x^2+4x-12x+8 +10x^2-9x-9=0 \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(x=-\frac{1}{18}x^2-\frac{9}{2} \\ \Leftrightarrow \frac{1}{18}x^2+x+\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2+x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-18}{2.1} & & \\ & = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{5}{8}x=-\frac{1}{4}x^2+\frac{9}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{5}{8}x-\frac{9}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{5}{8}x-\frac{9}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
11. \(x(x-6)=2(x-6) \\ \Leftrightarrow x^2-6x=2x-12 \\ \Leftrightarrow x^2-8x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.12 & &\\ & = 64-48 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt16}{2.1} & & = \frac{-(-8)+\sqrt16}{2.1} \\ & = \frac{4}{2} & & = \frac{12}{2} \\ & = 2 & & = 6 \\ \\ V &= \Big\{ 2 ; 6 \Big\} & &\end{align} \\ -----------------\)
12. \(x(9x-72)=-36(x+1) \\ \Leftrightarrow 9x^2-72x=-36x-36 \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)