# VKV met breuken of rekenwerk

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

1. $$-(7-38x)=-24x^2-(13-13x)$$
2. $$(x+1)(3x+3)-x(2x-1)=47$$
3. $$-(10-39x)=-2x^2-(82-14x)$$
4. $$(-3x+2)(3x+4)-x(-10x+18)=-113$$
5. $$-(13+x)=-x^2-(62-13x)$$
6. $$(x-3)(4x+4)-x(-12x-64)=-61$$
7. $$\frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}=0$$
8. $$\frac{1}{2}x^2-5x-\frac{11}{2}=0$$
9. $$(-x-5)(2x-4)-x(-5x+19)=32$$
10. $$2x^2-(13x+27)=x(x-19)$$
11. $$x(x+26)=5(x-22)$$
12. $$\frac{17}{10}x=-\frac{1}{5}x^2-\frac{4}{5}$$

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

##### Verbetersleutel

1. -(7-38x)=-24x^2-(13-13x) \\ \Leftrightarrow -7+38x=-24x^2-13+13x \\ \Leftrightarrow 24x^2+25x+6=0 \\\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} \\ -----------------
2. (x+1)(3x+3)-x(2x-1)=47\\ \Leftrightarrow 3x^2+3x+3x+3 -2x^2+x-47=0 \\ \Leftrightarrow x^2+7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-44=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.1.(-44) & &\\ & = 49+176 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt225}{2.1} & & = \frac{-7+\sqrt225}{2.1} \\ & = \frac{-22}{2} & & = \frac{8}{2} \\ & = -11 & & = 4 \\ \\ V &= \Big\{ -11 ; 4 \Big\} & &\end{align} \\ -----------------
3. -(10-39x)=-2x^2-(82-14x) \\ \Leftrightarrow -10+39x=-2x^2-82+14x \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------
4. (-3x+2)(3x+4)-x(-10x+18)=-113\\ \Leftrightarrow -9x^2-12x+6x+8 +10x^2-18x+113=0 \\ \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} \\ -----------------
5. -(13+x)=-x^2-(62-13x) \\ \Leftrightarrow -13-x=-x^2-62+13x \\ \Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-14)}{2.1} & & \\ & = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------
6. (x-3)(4x+4)-x(-12x-64)=-61\\ \Leftrightarrow 4x^2+4x-12x-12 +12x^2+64x+61=0 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------
7. \frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}\right)=0 \color{red}{.18} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{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. \frac{1}{2}x^2-5x-\frac{11}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-5x-\frac{11}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow x^2-10x-11=0 \\\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} \\ -----------------
9. (-x-5)(2x-4)-x(-5x+19)=32\\ \Leftrightarrow -2x^2+4x-10x+20 +5x^2-19x-32=0 \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------
10. 2x^2-(13x+27)=x(x-19) \\ \Leftrightarrow 2x^2-13x-27=x^2-19x \\ \Leftrightarrow x^2+6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-27=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.(-27) & &\\ & = 36+108 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt144}{2.1} & & = \frac{-6+\sqrt144}{2.1} \\ & = \frac{-18}{2} & & = \frac{6}{2} \\ & = -9 & & = 3 \\ \\ V &= \Big\{ -9 ; 3 \Big\} & &\end{align} \\ -----------------
11. x(x+26)=5(x-22) \\ \Leftrightarrow x^2+26x=5x-110 \\ \Leftrightarrow x^2+21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-21-\sqrt1}{2.1} & & = \frac{-21+\sqrt1}{2.1} \\ & = \frac{-22}{2} & & = \frac{-20}{2} \\ & = -11 & & = -10 \\ \\ V &= \Big\{ -11 ; -10 \Big\} & &\end{align} \\ -----------------
12. \frac{17}{10}x=-\frac{1}{5}x^2-\frac{4}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{17}{10}x+\frac{4}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{17}{10}x+\frac{4}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------
Oefeningengenerator vanhoeckes.be/wiskunde 2022-06-27 07:54:20