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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(4x+20)=4(x-4)\)
2. \(-\frac{5}{4}x=-2x^2-\frac{9}{8}\)
3. \(-\frac{7}{2}x=-x^2-\frac{49}{16}\)
4. \(-(11-19x)=-8x^2-(-7-12x)\)
5. \(5x^2-(19x-10)=4x(x-2)\)
6. \(x(9x-7)=-49(x+1)\)
7. \(\frac{1}{5}x^2+\frac{4}{5}x-\frac{77}{5}=0\)
8. \(-(11-42x)=-x^2-(132-20x)\)
9. \(x(18x+17)=-8(x+1)\)
10. \(\frac{1}{4}x^2+\frac{13}{12}x+1=0\)
11. \(x(48x+10)=3(x+1)\)
12. \(3x^2-(15x+18)=2x(x-6)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(4x+20)=4(x-4) \\ \Leftrightarrow 4x^2+20x=4x-16 \\ \Leftrightarrow 4x^2+16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
2. \(-\frac{5}{4}x=-2x^2-\frac{9}{8} \\ \Leftrightarrow 2x^2-\frac{5}{4}x+\frac{9}{8}=0 \\ \Leftrightarrow \color{red}{8.} \left(2x^2-\frac{5}{4}x+\frac{9}{8}\right)=0 \color{red}{.8} \\ \Leftrightarrow 16x^2-10x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-10x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.16.9 & &\\ & = 100-576 & & \\ & = -476 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(-\frac{7}{2}x=-x^2-\frac{49}{16} \\ \Leftrightarrow x^2-\frac{7}{2}x+\frac{49}{16}=0 \\ \Leftrightarrow \color{red}{16.} \left(x^2-\frac{7}{2}x+\frac{49}{16}\right)=0 \color{red}{.16} \\ \Leftrightarrow 16x^2-56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-56)}{2.16} & & \\ & = \frac{7}{4} & & \\V &= \Big\{ \frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(-(11-19x)=-8x^2-(-7-12x) \\ \Leftrightarrow -11+19x=-8x^2+7+12x \\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
5. \(5x^2-(19x-10)=4x(x-2) \\ \Leftrightarrow 5x^2-19x+10=4x^2-8x \\ \Leftrightarrow x^2-11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\ & = \frac{2}{2} & & = \frac{20}{2} \\ & = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
6. \(x(9x-7)=-49(x+1) \\ \Leftrightarrow 9x^2-7x=-49x-49 \\ \Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-42}{2.9} & & \\ & = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{5}x^2+\frac{4}{5}x-\frac{77}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{4}{5}x-\frac{77}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+4x-77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-77=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-77) & &\\ & = 16+308 & & \\ & = 324 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt324}{2.1} & & = \frac{-4+\sqrt324}{2.1} \\ & = \frac{-22}{2} & & = \frac{14}{2} \\ & = -11 & & = 7 \\ \\ V &= \Big\{ -11 ; 7 \Big\} & &\end{align} \\ -----------------\)
8. \(-(11-42x)=-x^2-(132-20x) \\ \Leftrightarrow -11+42x=-x^2-132+20x \\ \Leftrightarrow x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-22}{2.1} & & \\ & = -11 & & \\V &= \Big\{ -11 \Big\} & &\end{align} \\ -----------------\)
9. \(x(18x+17)=-8(x+1) \\ \Leftrightarrow 18x^2+17x=-8x-8 \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{1}{4}x^2+\frac{13}{12}x+1=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{4}x^2+\frac{13}{12}x+1\right)=0 \color{red}{.12} \\ \Leftrightarrow 3x^2+13x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+13x+12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.3.12 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.3} & & = \frac{-13+\sqrt25}{2.3} \\ & = \frac{-18}{6} & & = \frac{-8}{6} \\ & = -3 & & = \frac{-4}{3} \\ \\ V &= \Big\{ -3 ; \frac{-4}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(x(48x+10)=3(x+1) \\ \Leftrightarrow 48x^2+10x=3x+3 \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
12. \(3x^2-(15x+18)=2x(x-6) \\ \Leftrightarrow 3x^2-15x-18=2x^2-12x \\ \Leftrightarrow x^2-3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-18) & &\\ & = 9+72 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt81}{2.1} & & = \frac{-(-3)+\sqrt81}{2.1} \\ & = \frac{-6}{2} & & = \frac{12}{2} \\ & = -3 & & = 6 \\ \\ V &= \Big\{ -3 ; 6 \Big\} & &\end{align} \\ -----------------\)