Vierkantsvergelijkingen (VKV)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(16x^2-26x+64=0\)
2. \(8x^2+13x+26=-12x+8\)
3. \(16x^2+17x+1=0\)
4. \(x^2-10x-11=0\)
5. \(16x^2-8x+1=0\)
6. \(x^2-6x-10=-7x-4\)
7. \(48x^2+7x-3=0\)
8. \(x^2-4x-32=0\)
9. \(x^2-14x+33=-4x+9\)
10. \(9x^2-36x+36=0\)
11. \(24x^2+25x+6=0\)
12. \(x^2-3x+20=9x-12\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-26x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-26)^2-4.16.64 & &\\ & = 676-4096 & & \\ & = -3420 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(8x^2+13x+26=-12x+8\\ \Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.8.18 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\ & = \frac{-32}{16} & & = \frac{-18}{16} \\ & = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-11) & &\\ & = 100+44 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt144}{2.1} & & = \frac{-(-10)+\sqrt144}{2.1} \\ & = \frac{-2}{2} & & = \frac{22}{2} \\ & = -1 & & = 11 \\ \\ V &= \Big\{ -1 ; 11 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(x^2-6x-10=-7x-4\\ \Leftrightarrow x^2+x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-6) & &\\ & = 1+24 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt25}{2.1} & & = \frac{-1+\sqrt25}{2.1} \\ & = \frac{-6}{2} & & = \frac{4}{2} \\ & = -3 & & = 2 \\ \\ V &= \Big\{ -3 ; 2 \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt144}{2.1} & & = \frac{-(-4)+\sqrt144}{2.1} \\ & = \frac{-8}{2} & & = \frac{16}{2} \\ & = -4 & & = 8 \\ \\ V &= \Big\{ -4 ; 8 \Big\} & &\end{align} \\ -----------------\)
9. \(x^2-14x+33=-4x+9\\ \Leftrightarrow x^2-10x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.24 & &\\ & = 100-96 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt4}{2.1} & & = \frac{-(-10)+\sqrt4}{2.1} \\ & = \frac{8}{2} & & = \frac{12}{2} \\ & = 4 & & = 6 \\ \\ V &= \Big\{ 4 ; 6 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
11. \(\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(x^2-3x+20=9x-12\\ \Leftrightarrow x^2-12x+32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.32 & &\\ & = 144-128 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt16}{2.1} & & = \frac{-(-12)+\sqrt16}{2.1} \\ & = \frac{8}{2} & & = \frac{16}{2} \\ & = 4 & & = 8 \\ \\ V &= \Big\{ 4 ; 8 \Big\} & &\end{align} \\ -----------------\)