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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(11+45x)=-16x^2-(47-3x)\)
  2. \(6x^2-(7x-18)=4x(x-5)\)
  3. \(5x^2-(6x+36)=x(x-13)\)
  4. \(x(12x+28)=3(x-4)\)
  5. \((-5x-2)(5x-4)-x(-26x+36)=41\)
  6. \(-(9-27x)=-2x^2-(-63-20x)\)
  7. \(-\frac{5}{3}x=-\frac{4}{3}x^2-\frac{49}{12}\)
  8. \(2x^2-(6x+6)=x(x-11)\)
  9. \(x(x+24)=2(x-60)\)
  10. \((2x+3)(-4x-4)-x(-17x+28)=-76\)
  11. \(\frac{3}{2}x=-\frac{1}{2}x^2-\frac{9}{8}\)
  12. \(\frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}=0\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(11+45x)=-16x^2-(47-3x) \\ \Leftrightarrow -11-45x=-16x^2-47+3x \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  2. \(6x^2-(7x-18)=4x(x-5) \\ \Leftrightarrow 6x^2-7x+18=4x^2-20x \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
  3. \(5x^2-(6x+36)=x(x-13) \\ \Leftrightarrow 5x^2-6x-36=x^2-13x \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
  4. \(x(12x+28)=3(x-4) \\ \Leftrightarrow 12x^2+28x=3x-12 \\ \Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.12.12 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\ & = \frac{-32}{24} & & = \frac{-18}{24} \\ & = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
  5. \((-5x-2)(5x-4)-x(-26x+36)=41\\ \Leftrightarrow -25x^2+20x-10x+8 +26x^2-36x-41=0 \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
  6. \(-(9-27x)=-2x^2-(-63-20x) \\ \Leftrightarrow -9+27x=-2x^2+63+20x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
  7. \(-\frac{5}{3}x=-\frac{4}{3}x^2-\frac{49}{12} \\ \Leftrightarrow \frac{4}{3}x^2-\frac{5}{3}x+\frac{49}{12}=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{4}{3}x^2-\frac{5}{3}x+\frac{49}{12}\right)=0 \color{red}{.12} \\ \Leftrightarrow 16x^2-20x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-20x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.16.49 & &\\ & = 400-3136 & & \\ & = -2736 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  8. \(2x^2-(6x+6)=x(x-11) \\ \Leftrightarrow 2x^2-6x-6=x^2-11x \\ \Leftrightarrow x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-6) & &\\ & = 25+24 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt49}{2.1} & & = \frac{-5+\sqrt49}{2.1} \\ & = \frac{-12}{2} & & = \frac{2}{2} \\ & = -6 & & = 1 \\ \\ V &= \Big\{ -6 ; 1 \Big\} & &\end{align} \\ -----------------\)
  9. \(x(x+24)=2(x-60) \\ \Leftrightarrow x^2+24x=2x-120 \\ \Leftrightarrow x^2+22x+120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.120 & &\\ & = 484-480 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-22-\sqrt4}{2.1} & & = \frac{-22+\sqrt4}{2.1} \\ & = \frac{-24}{2} & & = \frac{-20}{2} \\ & = -12 & & = -10 \\ \\ V &= \Big\{ -12 ; -10 \Big\} & &\end{align} \\ -----------------\)
  10. \((2x+3)(-4x-4)-x(-17x+28)=-76\\ \Leftrightarrow -8x^2-8x-12x-12 +17x^2-28x+76=0 \\ \Leftrightarrow 9x^2-48x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-48x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.9.64 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.9} & & \\ & = \frac{8}{3} & & \\V &= \Big\{ \frac{8}{3} \Big\} & &\end{align} \\ -----------------\)
  11. \(\frac{3}{2}x=-\frac{1}{2}x^2-\frac{9}{8} \\ \Leftrightarrow \frac{1}{2}x^2+\frac{3}{2}x+\frac{9}{8}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{2}x^2+\frac{3}{2}x+\frac{9}{8}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
  12. \(\frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{5}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
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