Vierkantsvergelijkingen (VKV)

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x^2-21x-9=-12x+1\)
2. \(4x^2-6x+81=0\)
3. \(4x^2+4x+1=0\)
4. \(2x^2+20x+19=7x+1\)
5. \(x^2+12x+11=0\)
6. \(x^2+2x-3=0\)
7. \(x^2+26x+45=11x+1\)
8. \(4x^2+25x+36=0\)
9. \(4x^2+19x+45=-9x-4\)
10. \(2x^2-2x+2=-5x+4\)
11. \(x^2+2x+13=-4x+4\)
12. \(16x^2-46x+100=0\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x^2-21x-9=-12x+1\\ \Leftrightarrow x^2-9x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.(-10) & &\\ & = 81+40 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt121}{2.1} & & = \frac{-(-9)+\sqrt121}{2.1} \\ & = \frac{-2}{2} & & = \frac{20}{2} \\ & = -1 & & = 10 \\ \\ V &= \Big\{ -1 ; 10 \Big\} & &\end{align} \\ -----------------\)
2. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2-6x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.4.81 & &\\ & = 36-1296 & & \\ & = -1260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(2x^2+20x+19=7x+1\\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
5. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.11 & &\\ & = 144-44 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt100}{2.1} & & = \frac{-12+\sqrt100}{2.1} \\ & = \frac{-22}{2} & & = \frac{-2}{2} \\ & = -11 & & = -1 \\ \\ V &= \Big\{ -11 ; -1 \Big\} & &\end{align} \\ -----------------\)
6. \(\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt16}{2.1} & & = \frac{-2+\sqrt16}{2.1} \\ & = \frac{-6}{2} & & = \frac{2}{2} \\ & = -3 & & = 1 \\ \\ V &= \Big\{ -3 ; 1 \Big\} & &\end{align} \\ -----------------\)
7. \(x^2+26x+45=11x+1\\ \Leftrightarrow x^2+15x+44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.44 & &\\ & = 225-176 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt49}{2.1} & & = \frac{-15+\sqrt49}{2.1} \\ & = \frac{-22}{2} & & = \frac{-8}{2} \\ & = -11 & & = -4 \\ \\ V &= \Big\{ -11 ; -4 \Big\} & &\end{align} \\ -----------------\)
8. \(\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(4x^2+19x+45=-9x-4\\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(2x^2-2x+2=-5x+4\\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(x^2+2x+13=-4x+4\\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
12. \(\text{We zoeken de oplossingen van } \color{blue}{16x^2-46x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-46)^2-4.16.100 & &\\ & = 2116-6400 & & \\ & = -4284 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)