Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+19x+84=0\)
2. \(x^2-13x+40=0\)
3. \(2x^2+x+10=-4x+8\)
4. \(x^2+x-132=0\)
5. \(x^2-x-90=0\)
6. \(4x^2+19x+40=9x-9\)
7. \(x^2+2x-24=0\)
8. \(16x^2-5x+16=11x+12\)
9. \(8x^2+7x-18=0\)
10. \(4x^2-15x+7=-3x-2\)
11. \(x^2+24x+66=9x+10\)
12. \(x^2+0x+0=7x+8\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+84=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (19)^2-4.1.84 & &\\ & = 361-336 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-19-\sqrt25}{2.1} & & = \frac{-19+\sqrt25}{2.1} \\ & = \frac{-24}{2} & & = \frac{-14}{2} \\ & = -12 & & = -7 \\ \\ V &= \Big\{ -12 ; -7 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.40 & &\\ & = 169-160 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt9}{2.1} & & = \frac{-(-13)+\sqrt9}{2.1} \\ & = \frac{10}{2} & & = \frac{16}{2} \\ & = 5 & & = 8 \\ \\ V &= \Big\{ 5 ; 8 \Big\} & &\end{align} \\ -----------------\)
3. \(2x^2+x+10=-4x+8\\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+x-132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-132) & &\\ & = 1+528 & & \\ & = 529 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt529}{2.1} & & = \frac{-1+\sqrt529}{2.1} \\ & = \frac{-24}{2} & & = \frac{22}{2} \\ & = -12 & & = 11 \\ \\ V &= \Big\{ -12 ; 11 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-x-90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-90) & &\\ & = 1+360 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt361}{2.1} & & = \frac{-(-1)+\sqrt361}{2.1} \\ & = \frac{-18}{2} & & = \frac{20}{2} \\ & = -9 & & = 10 \\ \\ V &= \Big\{ -9 ; 10 \Big\} & &\end{align} \\ -----------------\)
6. \(4x^2+19x+40=9x-9\\ \Leftrightarrow 4x^2+10x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+10x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.4.49 & &\\ & = 100-784 & & \\ & = -684 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt100}{2.1} & & = \frac{-2+\sqrt100}{2.1} \\ & = \frac{-12}{2} & & = \frac{8}{2} \\ & = -6 & & = 4 \\ \\ V &= \Big\{ -6 ; 4 \Big\} & &\end{align} \\ -----------------\)
8. \(16x^2-5x+16=11x+12\\ \Leftrightarrow 16x^2-16x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.4 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.16} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
10. \(4x^2-15x+7=-3x-2\\ \Leftrightarrow 4x^2-12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.4} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(x^2+24x+66=9x+10\\ \Leftrightarrow x^2+15x+56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.56 & &\\ & = 225-224 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt1}{2.1} & & = \frac{-15+\sqrt1}{2.1} \\ & = \frac{-16}{2} & & = \frac{-14}{2} \\ & = -8 & & = -7 \\ \\ V &= \Big\{ -8 ; -7 \Big\} & &\end{align} \\ -----------------\)
12. \(x^2+0x+0=7x+8\\ \Leftrightarrow x^2-7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-8) & &\\ & = 49+32 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\ & = \frac{-2}{2} & & = \frac{16}{2} \\ & = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)