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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x+14)=24(x+1)\)
2. \(4x^2-(12x+70)=3x(x-5)\)
3. \(x(x+25)=33(x+1)\)
4. \(2x^2-(7x-7)=x(x-15)\)
5. \(25x^2-(13x-24)=19x(x-2)\)
6. \(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0\)
7. \((-3x+2)(2x-3)-x(-7x+9)=34\)
8. \((-3x+3)(-4x+2)-x(3x-5)=10\)
9. \(-(9-27x)=-x^2-(79-10x)\)
10. \(-(6-32x)=-4x^2-(31-12x)\)
11. \(\frac{1}{3}x^2+\frac{25}{6}x+12=0\)
12. \(x(x+12)=20(x+1)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x+14)=24(x+1) \\ \Leftrightarrow x^2+14x=24x+24 \\ \Leftrightarrow x^2-10x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt196}{2.1} & & = \frac{-(-10)+\sqrt196}{2.1} \\ & = \frac{-4}{2} & & = \frac{24}{2} \\ & = -2 & & = 12 \\ \\ V &= \Big\{ -2 ; 12 \Big\} & &\end{align} \\ -----------------\)
2. \(4x^2-(12x+70)=3x(x-5) \\ \Leftrightarrow 4x^2-12x-70=3x^2-15x \\ \Leftrightarrow x^2+3x-70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-70) & &\\ & = 9+280 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt289}{2.1} & & = \frac{-3+\sqrt289}{2.1} \\ & = \frac{-20}{2} & & = \frac{14}{2} \\ & = -10 & & = 7 \\ \\ V &= \Big\{ -10 ; 7 \Big\} & &\end{align} \\ -----------------\)
3. \(x(x+25)=33(x+1) \\ \Leftrightarrow x^2+25x=33x+33 \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
4. \(2x^2-(7x-7)=x(x-15) \\ \Leftrightarrow 2x^2-7x+7=x^2-15x \\ \Leftrightarrow x^2+8x+7=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+7=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.7 & &\\ & = 64-28 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt36}{2.1} & & = \frac{-8+\sqrt36}{2.1} \\ & = \frac{-14}{2} & & = \frac{-2}{2} \\ & = -7 & & = -1 \\ \\ V &= \Big\{ -7 ; -1 \Big\} & &\end{align} \\ -----------------\)
5. \(25x^2-(13x-24)=19x(x-2) \\ \Leftrightarrow 25x^2-13x+24=19x^2-38x \\ \Leftrightarrow 6x^2+25x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+25x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.6.24 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.6} & & = \frac{-25+\sqrt49}{2.6} \\ & = \frac{-32}{12} & & = \frac{-18}{12} \\ & = \frac{-8}{3} & & = \frac{-3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{-3}{2} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
7. \((-3x+2)(2x-3)-x(-7x+9)=34\\ \Leftrightarrow -6x^2+9x+4x-6 +7x^2-9x-34=0 \\ \Leftrightarrow x^2-6x-40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-40) & &\\ & = 36+160 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt196}{2.1} & & = \frac{-(-6)+\sqrt196}{2.1} \\ & = \frac{-8}{2} & & = \frac{20}{2} \\ & = -4 & & = 10 \\ \\ V &= \Big\{ -4 ; 10 \Big\} & &\end{align} \\ -----------------\)
8. \((-3x+3)(-4x+2)-x(3x-5)=10\\ \Leftrightarrow 12x^2-6x-12x+6 -3x^2+5x-10=0 \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
9. \(-(9-27x)=-x^2-(79-10x) \\ \Leftrightarrow -9+27x=-x^2-79+10x \\ \Leftrightarrow x^2+17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+70=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.70 & &\\ & = 289-280 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt9}{2.1} & & = \frac{-17+\sqrt9}{2.1} \\ & = \frac{-20}{2} & & = \frac{-14}{2} \\ & = -10 & & = -7 \\ \\ V &= \Big\{ -10 ; -7 \Big\} & &\end{align} \\ -----------------\)
10. \(-(6-32x)=-4x^2-(31-12x) \\ \Leftrightarrow -6+32x=-4x^2-31+12x \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{3}x^2+\frac{25}{6}x+12=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{25}{6}x+12\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(x(x+12)=20(x+1) \\ \Leftrightarrow x^2+12x=20x+20 \\ \Leftrightarrow x^2-8x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-20) & &\\ & = 64+80 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt144}{2.1} & & = \frac{-(-8)+\sqrt144}{2.1} \\ & = \frac{-4}{2} & & = \frac{20}{2} \\ & = -2 & & = 10 \\ \\ V &= \Big\{ -2 ; 10 \Big\} & &\end{align} \\ -----------------\)