# Vierkantsvergelijkingen (VKV)

#### Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. $$4x^2+17x-3=4x-12$$
2. $$x^2+15x+44=0$$
3. $$x^2-16x+55=0$$
4. $$x^2-16x+121=0$$
5. $$x^2+28x+125=6x+4$$
6. $$4x^2+17x+92=-x-8$$
7. $$16x^2+17x+1=0$$
8. $$4x^2+36x+81=0$$
9. $$4x^2+12x-42=5x-6$$
10. $$4x^2+11x-2=-6x-6$$
11. $$x^2-x-49=-8x+11$$
12. $$x^2-20x-14=-9x-2$$

#### Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

##### Verbetersleutel

1. 4x^2+17x-3=4x-12\\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------
2. \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} \\ -----------------
3. \text{We zoeken de oplossingen van } \color{blue}{x^2-16x+55=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.55 & &\\ & = 256-220 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt36}{2.1} & & = \frac{-(-16)+\sqrt36}{2.1} \\ & = \frac{10}{2} & & = \frac{22}{2} \\ & = 5 & & = 11 \\ \\ V &= \Big\{ 5 ; 11 \Big\} & &\end{align} \\ -----------------
4. \text{We zoeken de oplossingen van } \color{blue}{x^2-16x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.121 & &\\ & = 256-484 & & \\ & = -228 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------
5. x^2+28x+125=6x+4\\ \Leftrightarrow x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-22}{2.1} & & \\ & = -11 & & \\V &= \Big\{ -11 \Big\} & &\end{align} \\ -----------------
6. 4x^2+17x+92=-x-8\\ \Leftrightarrow 4x^2+18x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+18x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.4.100 & &\\ & = 324-1600 & & \\ & = -1276 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------
7. \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} \\ -----------------
8. \text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------
9. 4x^2+12x-42=5x-6\\ \Leftrightarrow 4x^2+7x-36=0 \\\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} \\ -----------------
10. 4x^2+11x-2=-6x-6\\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------
11. x^2-x-49=-8x+11\\ \Leftrightarrow x^2+7x-60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-60) & &\\ & = 49+240 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt289}{2.1} & & = \frac{-7+\sqrt289}{2.1} \\ & = \frac{-24}{2} & & = \frac{10}{2} \\ & = -12 & & = 5 \\ \\ V &= \Big\{ -12 ; 5 \Big\} & &\end{align} \\ -----------------
12. x^2-20x-14=-9x-2\\ \Leftrightarrow x^2-11x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.(-12) & &\\ & = 121+48 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt169}{2.1} & & = \frac{-(-11)+\sqrt169}{2.1} \\ & = \frac{-2}{2} & & = \frac{24}{2} \\ & = -1 & & = 12 \\ \\ V &= \Big\{ -1 ; 12 \Big\} & &\end{align} \\ -----------------
Oefeningengenerator vanhoeckes.be/wiskunde 2021-12-02 00:56:21