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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(36x+27)=2(x-2)\)
2. \(53x^2-(5x-3)=5x(x-6)\)
3. \(\frac{5}{6}x=-\frac{1}{3}x^2+3\)
4. \(7x^2-(5x+18)=6x(x-2)\)
5. \(-(5+6x)=-4x^2-(149-18x)\)
6. \((-4x-4)(-2x-1)-x(-8x-16)=-5\)
7. \(\frac{15}{16}x=-\frac{1}{4}x^2+\frac{1}{4}\)
8. \(5x^2-(19x-49)=x(x-47)\)
9. \((3x-3)(-4x+2)-x(-13x+5)=8\)
10. \(-(9+31x)=-16x^2-(34-9x)\)
11. \(-(11-2x)=-x^2-(26-10x)\)
12. \(-(10-32x)=-3x^2-(58-7x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(36x+27)=2(x-2) \\ \Leftrightarrow 36x^2+27x=2x-4 \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(53x^2-(5x-3)=5x(x-6) \\ \Leftrightarrow 53x^2-5x+3=5x^2-30x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{5}{6}x=-\frac{1}{3}x^2+3 \\ \Leftrightarrow \frac{1}{3}x^2+\frac{5}{6}x-3=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{3}x^2+\frac{5}{6}x-3\right)=0 \color{red}{.6} \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \(7x^2-(5x+18)=6x(x-2) \\ \Leftrightarrow 7x^2-5x-18=6x^2-12x \\ \Leftrightarrow x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-18) & &\\ & = 49+72 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt121}{2.1} & & = \frac{-7+\sqrt121}{2.1} \\ & = \frac{-18}{2} & & = \frac{4}{2} \\ & = -9 & & = 2 \\ \\ V &= \Big\{ -9 ; 2 \Big\} & &\end{align} \\ -----------------\)
5. \(-(5+6x)=-4x^2-(149-18x) \\ \Leftrightarrow -5-6x=-4x^2-149+18x \\ \Leftrightarrow 4x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.4.144 & &\\ & = 576-2304 & & \\ & = -1728 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \((-4x-4)(-2x-1)-x(-8x-16)=-5\\ \Leftrightarrow 8x^2+4x+8x+4 +8x^2+16x+5=0 \\ \Leftrightarrow 16x^2+24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+24x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.16.9 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.16} & & \\ & = -\frac{3}{4} & & \\V &= \Big\{ -\frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{15}{16}x=-\frac{1}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(5x^2-(19x-49)=x(x-47) \\ \Leftrightarrow 5x^2-19x+49=x^2-47x \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
9. \((3x-3)(-4x+2)-x(-13x+5)=8\\ \Leftrightarrow -12x^2+6x+12x-6 +13x^2-5x-8=0 \\ \Leftrightarrow x^2-5x-14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-14) & &\\ & = 25+56 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt81}{2.1} & & = \frac{-(-5)+\sqrt81}{2.1} \\ & = \frac{-4}{2} & & = \frac{14}{2} \\ & = -2 & & = 7 \\ \\ V &= \Big\{ -2 ; 7 \Big\} & &\end{align} \\ -----------------\)
10. \(-(9+31x)=-16x^2-(34-9x) \\ \Leftrightarrow -9-31x=-16x^2-34+9x \\ \Leftrightarrow 16x^2-40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.16} & & \\ & = \frac{5}{4} & & \\V &= \Big\{ \frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
11. \(-(11-2x)=-x^2-(26-10x) \\ \Leftrightarrow -11+2x=-x^2-26+10x \\ \Leftrightarrow x^2-8x+15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+15=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.15 & &\\ & = 64-60 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt4}{2.1} & & = \frac{-(-8)+\sqrt4}{2.1} \\ & = \frac{6}{2} & & = \frac{10}{2} \\ & = 3 & & = 5 \\ \\ V &= \Big\{ 3 ; 5 \Big\} & &\end{align} \\ -----------------\)
12. \(-(10-32x)=-3x^2-(58-7x) \\ \Leftrightarrow -10+32x=-3x^2-58+7x \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)