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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2-(18x+72)=x(x-17)\)
2. \(\frac{13}{2}x=-2x^2-\frac{9}{2}\)
3. \(7x^2-(16x+35)=6x(x-3)\)
4. \(-(10-42x)=-4x^2-(46-18x)\)
5. \(\frac{1}{16}x^2-\frac{1}{2}x+1=0\)
6. \((-4x+4)(-x-4)-x(3x+14)=-56\)
7. \(-\frac{13}{5}x=-\frac{1}{5}x^2-\frac{36}{5}\)
8. \(\frac{1}{5}x^2+3x+\frac{44}{5}=0\)
9. \(x(x+11)=12(x+1)\)
10. \(x(16x+18)=8(x-8)\)
11. \(\frac{1}{4}x^2+\frac{3}{4}x+\frac{9}{16}=0\)
12. \(5x^2-(17x-25)=x(x-37)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(2x^2-(18x+72)=x(x-17) \\ \Leftrightarrow 2x^2-18x-72=x^2-17x \\ \Leftrightarrow x^2-x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt289}{2.1} & & = \frac{-(-1)+\sqrt289}{2.1} \\ & = \frac{-16}{2} & & = \frac{18}{2} \\ & = -8 & & = 9 \\ \\ V &= \Big\{ -8 ; 9 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{13}{2}x=-2x^2-\frac{9}{2} \\ \Leftrightarrow 2x^2+\frac{13}{2}x+\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{2.} \left(2x^2+\frac{13}{2}x+\frac{9}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
3. \(7x^2-(16x+35)=6x(x-3) \\ \Leftrightarrow 7x^2-16x-35=6x^2-18x \\ \Leftrightarrow x^2+2x-35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-35) & &\\ & = 4+140 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt144}{2.1} & & = \frac{-2+\sqrt144}{2.1} \\ & = \frac{-14}{2} & & = \frac{10}{2} \\ & = -7 & & = 5 \\ \\ V &= \Big\{ -7 ; 5 \Big\} & &\end{align} \\ -----------------\)
4. \(-(10-42x)=-4x^2-(46-18x) \\ \Leftrightarrow -10+42x=-4x^2-46+18x \\ \Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.4.36 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.4} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{16}x^2-\frac{1}{2}x+1=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{2}x+1\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-32)}{2.4} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
6. \((-4x+4)(-x-4)-x(3x+14)=-56\\ \Leftrightarrow 4x^2+16x-4x-16 -3x^2-14x+56=0 \\ \Leftrightarrow x^2-14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
7. \(-\frac{13}{5}x=-\frac{1}{5}x^2-\frac{36}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{13}{5}x+\frac{36}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{13}{5}x+\frac{36}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-13x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.36 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt25}{2.1} & & = \frac{-(-13)+\sqrt25}{2.1} \\ & = \frac{8}{2} & & = \frac{18}{2} \\ & = 4 & & = 9 \\ \\ V &= \Big\{ 4 ; 9 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+3x+\frac{44}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+3x+\frac{44}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+15x+44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt49}{2.1} & & = \frac{-15+\sqrt49}{2.1} \\ & = \frac{-22}{2} & & = \frac{-8}{2} \\ & = -11 & & = -4 \\ \\ V &= \Big\{ -11 ; -4 \Big\} & &\end{align} \\ -----------------\)
9. \(x(x+11)=12(x+1) \\ \Leftrightarrow x^2+11x=12x+12 \\ \Leftrightarrow x^2-x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-12) & &\\ & = 1+48 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt49}{2.1} & & = \frac{-(-1)+\sqrt49}{2.1} \\ & = \frac{-6}{2} & & = \frac{8}{2} \\ & = -3 & & = 4 \\ \\ V &= \Big\{ -3 ; 4 \Big\} & &\end{align} \\ -----------------\)
10. \(x(16x+18)=8(x-8) \\ \Leftrightarrow 16x^2+18x=8x-64 \\ \Leftrightarrow 16x^2+10x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.64 & &\\ & = 100-4096 & & \\ & = -3996 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(\frac{1}{4}x^2+\frac{3}{4}x+\frac{9}{16}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2+\frac{3}{4}x+\frac{9}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(5x^2-(17x-25)=x(x-37) \\ \Leftrightarrow 5x^2-17x+25=x^2-37x \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)