Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+4x-21=0\)
2. \(9x^2+12x+49=0\)
3. \(16x^2+6x-1=0\)
4. \(18x^2+25x+8=0\)
5. \(x^2-5x+4=x-5\)
6. \(x^2-9x+132=-3x-12\)
7. \(4x^2-40x+100=0\)
8. \(4x^2-7x+101=-9x+1\)
9. \(x^2+5x+6=0\)
10. \(x^2-14x+49=0\)
11. \(4x^2+5x-6=-12x-10\)
12. \(x^2-8x+13=-x+7\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt100}{2.1} & & = \frac{-4+\sqrt100}{2.1} \\ & = \frac{-14}{2} & & = \frac{6}{2} \\ & = -7 & & = 3 \\ \\ V &= \Big\{ -7 ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2+12x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.9.49 & &\\ & = 144-1764 & & \\ & = -1620 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(x^2-5x+4=x-5\\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
6. \(x^2-9x+132=-3x-12\\ \Leftrightarrow x^2-6x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.144 & &\\ & = 36-576 & & \\ & = -540 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-40x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.4.100 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-40)}{2.4} & & \\ & = 5 & & \\V &= \Big\{ 5 \Big\} & &\end{align} \\ -----------------\)
8. \(4x^2-7x+101=-9x+1\\ \Leftrightarrow 4x^2+2x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.100 & &\\ & = 4-1600 & & \\ & = -1596 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt1}{2.1} & & = \frac{-5+\sqrt1}{2.1} \\ & = \frac{-6}{2} & & = \frac{-4}{2} \\ & = -3 & & = -2 \\ \\ V &= \Big\{ -3 ; -2 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
11. \(4x^2+5x-6=-12x-10\\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
12. \(x^2-8x+13=-x+7\\ \Leftrightarrow x^2-7x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.6 & &\\ & = 49-24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt25}{2.1} & & = \frac{-(-7)+\sqrt25}{2.1} \\ & = \frac{2}{2} & & = \frac{12}{2} \\ & = 1 & & = 6 \\ \\ V &= \Big\{ 1 ; 6 \Big\} & &\end{align} \\ -----------------\)