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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

  1. \(-(2-23x)=-4x^2-(6-6x)\)
  2. \(-(13-41x)=-9x^2-(77-13x)\)
  3. \(\frac{3}{5}x=-\frac{8}{5}x^2+\frac{1}{10}\)
  4. \(2x^2-(9x+12)=x(x-5)\)
  5. \(10x^2-(2x-49)=x(x+22)\)
  6. \((-4x+4)(5x+5)-x(-44x-7)=26\)
  7. \(2x^2-(3x+40)=x(x+3)\)
  8. \((2x+3)(-x+5)-x(-3x+24)=71\)
  9. \((4x+5)(x-3)-x(3x-16)=-43\)
  10. \(x^2+\frac{15}{8}x-\frac{1}{4}=0\)
  11. \(-(15-18x)=-4x^2-(40-12x)\)
  12. \(x(x+50)=48(x+1)\)

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

  1. \(-(2-23x)=-4x^2-(6-6x) \\ \Leftrightarrow -2+23x=-4x^2-6+6x \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
  2. \(-(13-41x)=-9x^2-(77-13x) \\ \Leftrightarrow -13+41x=-9x^2-77+13x \\ \Leftrightarrow 9x^2+28x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+28x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.9.64 & &\\ & = 784-2304 & & \\ & = -1520 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  3. \(\frac{3}{5}x=-\frac{8}{5}x^2+\frac{1}{10} \\ \Leftrightarrow \frac{8}{5}x^2+\frac{3}{5}x-\frac{1}{10}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{8}{5}x^2+\frac{3}{5}x-\frac{1}{10}\right)=0 \color{red}{.10} \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  4. \(2x^2-(9x+12)=x(x-5) \\ \Leftrightarrow 2x^2-9x-12=x^2-5x \\ \Leftrightarrow x^2-4x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-12) & &\\ & = 16+48 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt64}{2.1} & & = \frac{-(-4)+\sqrt64}{2.1} \\ & = \frac{-4}{2} & & = \frac{12}{2} \\ & = -2 & & = 6 \\ \\ V &= \Big\{ -2 ; 6 \Big\} & &\end{align} \\ -----------------\)
  5. \(10x^2-(2x-49)=x(x+22) \\ \Leftrightarrow 10x^2-2x+49=x^2+22x \\ \Leftrightarrow 9x^2-24x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.49 & &\\ & = 576-1764 & & \\ & = -1188 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  6. \((-4x+4)(5x+5)-x(-44x-7)=26\\ \Leftrightarrow -20x^2-20x+20x+20 +44x^2+7x-26=0 \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
  7. \(2x^2-(3x+40)=x(x+3) \\ \Leftrightarrow 2x^2-3x-40=x^2+3x \\ \Leftrightarrow x^2-6x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-40) & &\\ & = 36+160 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt196}{2.1} & & = \frac{-(-6)+\sqrt196}{2.1} \\ & = \frac{-8}{2} & & = \frac{20}{2} \\ & = -4 & & = 10 \\ \\ V &= \Big\{ -4 ; 10 \Big\} & &\end{align} \\ -----------------\)
  8. \((2x+3)(-x+5)-x(-3x+24)=71\\ \Leftrightarrow -2x^2+10x-3x+15 +3x^2-24x-71=0 \\ \Leftrightarrow x^2+x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt225}{2.1} & & = \frac{-1+\sqrt225}{2.1} \\ & = \frac{-16}{2} & & = \frac{14}{2} \\ & = -8 & & = 7 \\ \\ V &= \Big\{ -8 ; 7 \Big\} & &\end{align} \\ -----------------\)
  9. \((4x+5)(x-3)-x(3x-16)=-43\\ \Leftrightarrow 4x^2-12x+5x-15 -3x^2+16x+43=0 \\ \Leftrightarrow x^2-11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt9}{2.1} & & = \frac{-(-11)+\sqrt9}{2.1} \\ & = \frac{8}{2} & & = \frac{14}{2} \\ & = 4 & & = 7 \\ \\ V &= \Big\{ 4 ; 7 \Big\} & &\end{align} \\ -----------------\)
  10. \(x^2+\frac{15}{8}x-\frac{1}{4}=0\\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{15}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
  11. \(-(15-18x)=-4x^2-(40-12x) \\ \Leftrightarrow -15+18x=-4x^2-40+12x \\ \Leftrightarrow 4x^2+6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+6x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.4.25 & &\\ & = 36-400 & & \\ & = -364 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
  12. \(x(x+50)=48(x+1) \\ \Leftrightarrow x^2+50x=48x+48 \\ \Leftrightarrow x^2+2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt196}{2.1} & & = \frac{-2+\sqrt196}{2.1} \\ & = \frac{-16}{2} & & = \frac{12}{2} \\ & = -8 & & = 6 \\ \\ V &= \Big\{ -8 ; 6 \Big\} & &\end{align} \\ -----------------\)
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