Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(3x^2+x-46=-6x+2\)
2. \(x^2+6x-30=9x-2\)
3. \(2x^2-3x-77=-10x-5\)
4. \(x^2-13x+28=-9x-8\)
5. \(x^2-12x+14=-2x-11\)
6. \(24x^2+7x-6=0\)
7. \(x^2+5x-35=10x-11\)
8. \(x^2-3x-88=0\)
9. \(x^2-6x-55=0\)
10. \(9x^2+42x+49=0\)
11. \(36x^2+7x-4=0\)
12. \(x^2+x+1=3x+9\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(3x^2+x-46=-6x+2\\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(x^2+6x-30=9x-2\\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
3. \(2x^2-3x-77=-10x-5\\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(x^2-13x+28=-9x-8\\ \Leftrightarrow x^2-4x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.36 & &\\ & = 16-144 & & \\ & = -128 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x^2-12x+14=-2x-11\\ \Leftrightarrow x^2-10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-10)}{2.1} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(x^2+5x-35=10x-11\\ \Leftrightarrow x^2-5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-24) & &\\ & = 25+96 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt121}{2.1} & & = \frac{-(-5)+\sqrt121}{2.1} \\ & = \frac{-6}{2} & & = \frac{16}{2} \\ & = -3 & & = 8 \\ \\ V &= \Big\{ -3 ; 8 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-88=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-88) & &\\ & = 9+352 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt361}{2.1} & & = \frac{-(-3)+\sqrt361}{2.1} \\ & = \frac{-16}{2} & & = \frac{22}{2} \\ & = -8 & & = 11 \\ \\ V &= \Big\{ -8 ; 11 \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.(-55) & &\\ & = 36+220 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt256}{2.1} & & = \frac{-(-6)+\sqrt256}{2.1} \\ & = \frac{-10}{2} & & = \frac{22}{2} \\ & = -5 & & = 11 \\ \\ V &= \Big\{ -5 ; 11 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (42)^2-4.9.49 & &\\ & = 1764-1764 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-42}{2.9} & & \\ & = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
12. \(x^2+x+1=3x+9\\ \Leftrightarrow x^2-2x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-8) & &\\ & = 4+32 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt36}{2.1} & & = \frac{-(-2)+\sqrt36}{2.1} \\ & = \frac{-4}{2} & & = \frac{8}{2} \\ & = -2 & & = 4 \\ \\ V &= \Big\{ -2 ; 4 \Big\} & &\end{align} \\ -----------------\)