Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+22x+50=6x-5\)
2. \(4x^2-44x+121=0\)
3. \(72x^2-5x+5=-12x+7\)
4. \(48x^2+7x-3=0\)
5. \(72x^2+19x+9=-6x+7\)
6. \(x^2-12x+32=0\)
7. \(9x^2+76x+139=4x-5\)
8. \(x^2+0x+2=9x-6\)
9. \(2x^2+7x-72=0\)
10. \(x^2-14x+40=0\)
11. \(x^2+11x-18=8x+10\)
12. \(4x^2+30x+72=-2x-9\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2+22x+50=6x-5\\ \Leftrightarrow x^2+16x+55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.55 & &\\ & = 256-220 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt36}{2.1} & & = \frac{-16+\sqrt36}{2.1} \\ & = \frac{-22}{2} & & = \frac{-10}{2} \\ & = -11 & & = -5 \\ \\ V &= \Big\{ -11 ; -5 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(72x^2-5x+5=-12x+7\\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
5. \(72x^2+19x+9=-6x+7\\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt16}{2.1} & & = \frac{-(-12)+\sqrt16}{2.1} \\ & = \frac{8}{2} & & = \frac{16}{2} \\ & = 4 & & = 8 \\ \\ V &= \Big\{ 4 ; 8 \Big\} & &\end{align} \\ -----------------\)
7. \(9x^2+76x+139=4x-5\\ \Leftrightarrow 9x^2+72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.9} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
8. \(x^2+0x+2=9x-6\\ \Leftrightarrow x^2-9x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.8 & &\\ & = 81-32 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt49}{2.1} & & = \frac{-(-9)+\sqrt49}{2.1} \\ & = \frac{2}{2} & & = \frac{16}{2} \\ & = 1 & & = 8 \\ \\ V &= \Big\{ 1 ; 8 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
11. \(x^2+11x-18=8x+10\\ \Leftrightarrow x^2+3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt121}{2.1} & & = \frac{-3+\sqrt121}{2.1} \\ & = \frac{-14}{2} & & = \frac{8}{2} \\ & = -7 & & = 4 \\ \\ V &= \Big\{ -7 ; 4 \Big\} & &\end{align} \\ -----------------\)
12. \(4x^2+30x+72=-2x-9\\ \Leftrightarrow 4x^2+32x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+32x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.4.81 & &\\ & = 1024-1296 & & \\ & = -272 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)