Vierkantsvergelijkingen (VKV)

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2+16x-39=11x-3\)
2. \(x^2+9x+10=x-6\)
3. \(12x^2+5x-13=-2x-1\)
4. \(4x^2+7x-36=0\)
5. \(9x^2-17x+11=-11x+10\)
6. \(4x^2+4x+1=0\)
7. \(6x^2+13x+6=0\)
8. \(9x^2-14x+24=10x+8\)
9. \(4x^2+40x+63=8x-1\)
10. \(4x^2+15x-4=0\)
11. \(x^2+12x+31=2x+10\)
12. \(4x^2-8x+121=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2+16x-39=11x-3\\ \Leftrightarrow x^2+5x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-36) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.1} & & = \frac{-5+\sqrt169}{2.1} \\ & = \frac{-18}{2} & & = \frac{8}{2} \\ & = -9 & & = 4 \\ \\ V &= \Big\{ -9 ; 4 \Big\} & &\end{align} \\ -----------------\)
2. \(x^2+9x+10=x-6\\ \Leftrightarrow x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.1} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
3. \(12x^2+5x-13=-2x-1\\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
5. \(9x^2-17x+11=-11x+10\\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
8. \(9x^2-14x+24=10x+8\\ \Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.9.16 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.9} & & \\ & = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(4x^2+40x+63=8x-1\\ \Leftrightarrow 4x^2+32x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+32x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.4.64 & &\\ & = 1024-1024 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-32}{2.4} & & \\ & = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)
10. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
11. \(x^2+12x+31=2x+10\\ \Leftrightarrow x^2+10x+21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.21 & &\\ & = 100-84 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt16}{2.1} & & = \frac{-10+\sqrt16}{2.1} \\ & = \frac{-14}{2} & & = \frac{-6}{2} \\ & = -7 & & = -3 \\ \\ V &= \Big\{ -7 ; -3 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-8x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.4.121 & &\\ & = 64-1936 & & \\ & = -1872 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)