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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{7}{9}x=-\frac{16}{3}x^2+\frac{1}{3}\)
2. \(x(16x+54)=6(x-6)\)
3. \(-(13-108x)=-16x^2-(134-20x)\)
4. \(-(13-5x)=-x^2-(17-10x)\)
5. \(10x^2-(14x+48)=7x(x-3)\)
6. \((-2x+2)(-4x+1)-x(7x+5)=86\)
7. \((x+3)(4x+5)-x(0x+7)=6\)
8. \(-(8-10x)=-2x^2-(6-7x)\)
9. \(-(7-7x)=-2x^2-(9-2x)\)
10. \(-(7+76x)=-16x^2-(128-12x)\)
11. \(-(13-39x)=-48x^2-(16-14x)\)
12. \(-(12-49x)=-16x^2-(37-9x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{7}{9}x=-\frac{16}{3}x^2+\frac{1}{3} \\ \Leftrightarrow \frac{16}{3}x^2+\frac{7}{9}x-\frac{1}{3}=0 \\ \Leftrightarrow \color{red}{9.} \left(\frac{16}{3}x^2+\frac{7}{9}x-\frac{1}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(x(16x+54)=6(x-6) \\ \Leftrightarrow 16x^2+54x=6x-36 \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(-(13-108x)=-16x^2-(134-20x) \\ \Leftrightarrow -13+108x=-16x^2-134+20x \\ \Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-88}{2.16} & & \\ & = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(-(13-5x)=-x^2-(17-10x) \\ \Leftrightarrow -13+5x=-x^2-17+10x \\ \Leftrightarrow x^2-5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt9}{2.1} & & = \frac{-(-5)+\sqrt9}{2.1} \\ & = \frac{2}{2} & & = \frac{8}{2} \\ & = 1 & & = 4 \\ \\ V &= \Big\{ 1 ; 4 \Big\} & &\end{align} \\ -----------------\)
5. \(10x^2-(14x+48)=7x(x-3) \\ \Leftrightarrow 10x^2-14x-48=7x^2-21x \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
6. \((-2x+2)(-4x+1)-x(7x+5)=86\\ \Leftrightarrow 8x^2-2x-8x+2 -7x^2-5x-86=0 \\ \Leftrightarrow x^2-5x-84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-84) & &\\ & = 25+336 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt361}{2.1} & & = \frac{-(-5)+\sqrt361}{2.1} \\ & = \frac{-14}{2} & & = \frac{24}{2} \\ & = -7 & & = 12 \\ \\ V &= \Big\{ -7 ; 12 \Big\} & &\end{align} \\ -----------------\)
7. \((x+3)(4x+5)-x(0x+7)=6\\ \Leftrightarrow 4x^2+5x+12x+15 +0x^2-7x-6=0 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
8. \(-(8-10x)=-2x^2-(6-7x) \\ \Leftrightarrow -8+10x=-2x^2-6+7x \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(-(7-7x)=-2x^2-(9-2x) \\ \Leftrightarrow -7+7x=-2x^2-9+2x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(-(7+76x)=-16x^2-(128-12x) \\ \Leftrightarrow -7-76x=-16x^2-128+12x \\ \Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-88)^2-4.16.121 & &\\ & = 7744-7744 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-88)}{2.16} & & \\ & = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
11. \(-(13-39x)=-48x^2-(16-14x) \\ \Leftrightarrow -13+39x=-48x^2-16+14x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(-(12-49x)=-16x^2-(37-9x) \\ \Leftrightarrow -12+49x=-16x^2-37+9x \\ \Leftrightarrow 16x^2+40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.16} & & \\ & = -\frac{5}{4} & & \\V &= \Big\{ -\frac{5}{4} \Big\} & &\end{align} \\ -----------------\)