Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-14x+40=0\)
2. \(x^2-4x+100=0\)
3. \(9x^2+5x+143=-7x-1\)
4. \(x^2-2x-26=x+2\)
5. \(x^2-x+13=-7x+5\)
6. \(x^2+24x+144=0\)
7. \(x^2-4x+3=0\)
8. \(12x^2+13x+3=0\)
9. \(x^2-5x-61=-7x+2\)
10. \(x^2+2x-24=0\)
11. \(4x^2-2x+1=0\)
12. \(8x^2+15x-25=8x-7\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.100 & &\\ & = 16-400 & & \\ & = -384 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(9x^2+5x+143=-7x-1\\ \Leftrightarrow 9x^2+12x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+12x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.9.144 & &\\ & = 144-5184 & & \\ & = -5040 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(x^2-2x-26=x+2\\ \Leftrightarrow x^2-3x-28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-28) & &\\ & = 9+112 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt121}{2.1} & & = \frac{-(-3)+\sqrt121}{2.1} \\ & = \frac{-8}{2} & & = \frac{14}{2} \\ & = -4 & & = 7 \\ \\ V &= \Big\{ -4 ; 7 \Big\} & &\end{align} \\ -----------------\)
5. \(x^2-x+13=-7x+5\\ \Leftrightarrow x^2+6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt4}{2.1} & & = \frac{-6+\sqrt4}{2.1} \\ & = \frac{-8}{2} & & = \frac{-4}{2} \\ & = -4 & & = -2 \\ \\ V &= \Big\{ -4 ; -2 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-24}{2.1} & & \\ & = -12 & & \\V &= \Big\{ -12 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt4}{2.1} & & = \frac{-(-4)+\sqrt4}{2.1} \\ & = \frac{2}{2} & & = \frac{6}{2} \\ & = 1 & & = 3 \\ \\ V &= \Big\{ 1 ; 3 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(x^2-5x-61=-7x+2\\ \Leftrightarrow x^2+2x-63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-63) & &\\ & = 4+252 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt256}{2.1} & & = \frac{-2+\sqrt256}{2.1} \\ & = \frac{-18}{2} & & = \frac{14}{2} \\ & = -9 & & = 7 \\ \\ V &= \Big\{ -9 ; 7 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-24) & &\\ & = 4+96 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt100}{2.1} & & = \frac{-2+\sqrt100}{2.1} \\ & = \frac{-12}{2} & & = \frac{8}{2} \\ & = -6 & & = 4 \\ \\ V &= \Big\{ -6 ; 4 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.1 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(8x^2+15x-25=8x-7\\ \Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.8.(-18) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\ & = \frac{-32}{16} & & = \frac{18}{16} \\ & = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)