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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(5x^2-(8x-121)=x(x-52)\)
  2. \(\frac{8}{3}x=-\frac{1}{3}x^2+3\)
  3. \((2x+1)(-4x-5)-x(-24x-39)=-14\)
  4. \(\frac{1}{5}x^2-\frac{13}{5}x+\frac{36}{5}=0\)
  5. \(\frac{16}{15}x^2+\frac{4}{3}x+\frac{5}{3}=0\)
  6. \((-2x+4)(x-4)-x(-3x-6)=83\)
  7. \((-3x-3)(2x-5)-x(-8x+23)=87\)
  8. \(8x^2-(15x+9)=4x(x-5)\)
  9. \(-(4+76x)=-16x^2-(125-12x)\)
  10. \(7x^2-(14x+6)=x(x-19)\)
  11. \(x(x+10)=8(x+1)\)
  12. \(-(7+41x)=-16x^2-(43-7x)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(5x^2-(8x-121)=x(x-52) \\ \Leftrightarrow 5x^2-8x+121=x^2-52x \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(\frac{8}{3}x=-\frac{1}{3}x^2+3 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{8}{3}x-3=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{8}{3}x-3\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+8x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-9) & &\\ & = 64+36 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt100}{2.1} & & = \frac{-8+\sqrt100}{2.1} \\ & = \frac{-18}{2} & & = \frac{2}{2} \\ & = -9 & & = 1 \\ \\ V &= \Big\{ -9 ; 1 \Big\} & &\end{align} \\ -----------------\)
  3. \((2x+1)(-4x-5)-x(-24x-39)=-14\\ \Leftrightarrow -8x^2-10x-4x-5 +24x^2+39x+14=0 \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(\frac{1}{5}x^2-\frac{13}{5}x+\frac{36}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{13}{5}x+\frac{36}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-13x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt25}{2.1} & & = \frac{-(-13)+\sqrt25}{2.1} \\ & = \frac{8}{2} & & = \frac{18}{2} \\ & = 4 & & = 9 \\ \\ V &= \Big\{ 4 ; 9 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{16}{15}x^2+\frac{4}{3}x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{15.} \left(\frac{16}{15}x^2+\frac{4}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 16x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.16.25 & &\\ & = 400-1600 & & \\ & = -1200 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \((-2x+4)(x-4)-x(-3x-6)=83\\ \Leftrightarrow -2x^2+8x+4x-16 +3x^2+6x-83=0 \\ \Leftrightarrow x^2-2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt400}{2.1} & & = \frac{-(-2)+\sqrt400}{2.1} \\ & = \frac{-18}{2} & & = \frac{22}{2} \\ & = -9 & & = 11 \\ \\ V &= \Big\{ -9 ; 11 \Big\} & &\end{align} \\ -----------------\)
  7. \((-3x-3)(2x-5)-x(-8x+23)=87\\ \Leftrightarrow -6x^2+15x-6x+15 +8x^2-23x-87=0 \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  8. \(8x^2-(15x+9)=4x(x-5) \\ \Leftrightarrow 8x^2-15x-9=4x^2-20x \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
  9. \(-(4+76x)=-16x^2-(125-12x) \\ \Leftrightarrow -4-76x=-16x^2-125+12x \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
  10. \(7x^2-(14x+6)=x(x-19) \\ \Leftrightarrow 7x^2-14x-6=x^2-19x \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(x(x+10)=8(x+1) \\ \Leftrightarrow x^2+10x=8x+8 \\ \Leftrightarrow x^2+2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt36}{2.1} & & = \frac{-2+\sqrt36}{2.1} \\ & = \frac{-8}{2} & & = \frac{4}{2} \\ & = -4 & & = 2 \\ \\ V &= \Big\{ -4 ; 2 \Big\} & &\end{align} \\ -----------------\)
  12. \(-(7+41x)=-16x^2-(43-7x) \\ \Leftrightarrow -7-41x=-16x^2-43+7x \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
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