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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(7-3x)=-9x^2-(8-9x)\)
  2. \(10x^2-(12x-5)=9x(x-2)\)
  3. \(\frac{1}{4}x^2+\frac{9}{4}x-\frac{11}{2}=0\)
  4. \(x(x-15)=6(x-18)\)
  5. \(\frac{21}{4}x=-\frac{1}{4}x^2-\frac{55}{2}\)
  6. \((-x+4)(2x-2)-x(-14x-13)=4\)
  7. \(-2x=-\frac{1}{3}x^2+\frac{55}{3}\)
  8. \((-2x+5)(-4x-2)-x(4x-22)=-26\)
  9. \((-3x-4)(-5x+5)-x(14x-16)=-110\)
  10. \(x(x-6)=-4(x+1)\)
  11. \(\frac{1}{8}x^2+\frac{3}{4}x+1=0\)
  12. \(x(x+33)=27(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(7-3x)=-9x^2-(8-9x) \\ \Leftrightarrow -7+3x=-9x^2-8+9x \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
  2. \(10x^2-(12x-5)=9x(x-2) \\ \Leftrightarrow 10x^2-12x+5=9x^2-18x \\ \Leftrightarrow x^2+6x+5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.5 & &\\ & = 36-20 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt16}{2.1} & & = \frac{-6+\sqrt16}{2.1} \\ & = \frac{-10}{2} & & = \frac{-2}{2} \\ & = -5 & & = -1 \\ \\ V &= \Big\{ -5 ; -1 \Big\} & &\end{align} \\ -----------------\)
  3. \(\frac{1}{4}x^2+\frac{9}{4}x-\frac{11}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{9}{4}x-\frac{11}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+9x-22=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-22=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-22) & &\\ & = 81+88 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt169}{2.1} & & = \frac{-9+\sqrt169}{2.1} \\ & = \frac{-22}{2} & & = \frac{4}{2} \\ & = -11 & & = 2 \\ \\ V &= \Big\{ -11 ; 2 \Big\} & &\end{align} \\ -----------------\)
  4. \(x(x-15)=6(x-18) \\ \Leftrightarrow x^2-15x=6x-108 \\ \Leftrightarrow x^2-21x+108=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+108=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.108 & &\\ & = 441-432 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt9}{2.1} & & = \frac{-(-21)+\sqrt9}{2.1} \\ & = \frac{18}{2} & & = \frac{24}{2} \\ & = 9 & & = 12 \\ \\ V &= \Big\{ 9 ; 12 \Big\} & &\end{align} \\ -----------------\)
  5. \(\frac{21}{4}x=-\frac{1}{4}x^2-\frac{55}{2} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{21}{4}x+\frac{55}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{21}{4}x+\frac{55}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-21-\sqrt1}{2.1} & & = \frac{-21+\sqrt1}{2.1} \\ & = \frac{-22}{2} & & = \frac{-20}{2} \\ & = -11 & & = -10 \\ \\ V &= \Big\{ -11 ; -10 \Big\} & &\end{align} \\ -----------------\)
  6. \((-x+4)(2x-2)-x(-14x-13)=4\\ \Leftrightarrow -2x^2+2x+8x-8 +14x^2+13x-4=0 \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
  7. \(-2x=-\frac{1}{3}x^2+\frac{55}{3} \\ \Leftrightarrow \frac{1}{3}x^2-2x-\frac{55}{3}=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-2x-\frac{55}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-6x-55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-55) & &\\ & = 36+220 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt256}{2.1} & & = \frac{-(-6)+\sqrt256}{2.1} \\ & = \frac{-10}{2} & & = \frac{22}{2} \\ & = -5 & & = 11 \\ \\ V &= \Big\{ -5 ; 11 \Big\} & &\end{align} \\ -----------------\)
  8. \((-2x+5)(-4x-2)-x(4x-22)=-26\\ \Leftrightarrow 8x^2+4x-20x-10 -4x^2+22x+26=0 \\ \Leftrightarrow 4x^2+16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
  9. \((-3x-4)(-5x+5)-x(14x-16)=-110\\ \Leftrightarrow 15x^2-15x+20x-20 -14x^2+16x+110=0 \\ \Leftrightarrow x^2-19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-19)^2-4.1.90 & &\\ & = 361-360 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-19)-\sqrt1}{2.1} & & = \frac{-(-19)+\sqrt1}{2.1} \\ & = \frac{18}{2} & & = \frac{20}{2} \\ & = 9 & & = 10 \\ \\ V &= \Big\{ 9 ; 10 \Big\} & &\end{align} \\ -----------------\)
  10. \(x(x-6)=-4(x+1) \\ \Leftrightarrow x^2-6x=-4x-4 \\ \Leftrightarrow x^2-2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.4 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  11. \(\frac{1}{8}x^2+\frac{3}{4}x+1=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{3}{4}x+1\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt4}{2.1} & & = \frac{-6+\sqrt4}{2.1} \\ & = \frac{-8}{2} & & = \frac{-4}{2} \\ & = -4 & & = -2 \\ \\ V &= \Big\{ -4 ; -2 \Big\} & &\end{align} \\ -----------------\)
  12. \(x(x+33)=27(x+1) \\ \Leftrightarrow x^2+33x=27x+27 \\ \Leftrightarrow x^2+6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt144}{2.1} & & = \frac{-6+\sqrt144}{2.1} \\ & = \frac{-18}{2} & & = \frac{6}{2} \\ & = -9 & & = 3 \\ \\ V &= \Big\{ -9 ; 3 \Big\} & &\end{align} \\ -----------------\)
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