VKV met breuken of rekenwerk

Hoofdmenu Eentje per keer  🖨

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x+61)=66(x+1)\)
2. \(x(x+15)=5(x-5)\)
3. \(x(16x+7)=-49(x+1)\)
4. \(-(2-39x)=-4x^2-(38-14x)\)
5. \(\frac{11}{4}x=-\frac{1}{4}x^2-7\)
6. \(x(16x-9)=-(x+1)\)
7. \((-3x-3)(-5x+2)-x(14x-3)=-14\)
8. \(x(2x+79)=72(x+1)\)
9. \(-(14-10x)=-x^2-(18-8x)\)
10. \(x(x+20)=4(x-16)\)
11. \(-(13-15x)=-x^2-(29-5x)\)
12. \(-(2-x)=-x^2-(-54-2x)\)

---

Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x+61)=66(x+1) \\ \Leftrightarrow x^2+61x=66x+66 \\ \Leftrightarrow x^2-5x-66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-66) & &\\ & = 25+264 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt289}{2.1} & & = \frac{-(-5)+\sqrt289}{2.1} \\ & = \frac{-12}{2} & & = \frac{22}{2} \\ & = -6 & & = 11 \\ \\ V &= \Big\{ -6 ; 11 \Big\} & &\end{align} \\ -----------------\)
2. \(x(x+15)=5(x-5) \\ \Leftrightarrow x^2+15x=5x-25 \\ \Leftrightarrow x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-10}{2.1} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
3. \(x(16x+7)=-49(x+1) \\ \Leftrightarrow 16x^2+7x=-49x-49 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(-(2-39x)=-4x^2-(38-14x) \\ \Leftrightarrow -2+39x=-4x^2-38+14x \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{11}{4}x=-\frac{1}{4}x^2-7 \\ \Leftrightarrow \frac{1}{4}x^2+\frac{11}{4}x+7=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{11}{4}x+7\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+28=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.28 & &\\ & = 121-112 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt9}{2.1} & & = \frac{-11+\sqrt9}{2.1} \\ & = \frac{-14}{2} & & = \frac{-8}{2} \\ & = -7 & & = -4 \\ \\ V &= \Big\{ -7 ; -4 \Big\} & &\end{align} \\ -----------------\)
6. \(x(16x-9)=-(x+1) \\ \Leftrightarrow 16x^2-9x=-x-1 \\ \Leftrightarrow 16x^2-8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.16} & & \\ & = \frac{1}{4} & & \\V &= \Big\{ \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
7. \((-3x-3)(-5x+2)-x(14x-3)=-14\\ \Leftrightarrow 15x^2-6x+15x-6 -14x^2+3x+14=0 \\ \Leftrightarrow x^2-9x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.8 & &\\ & = 81-32 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt49}{2.1} & & = \frac{-(-9)+\sqrt49}{2.1} \\ & = \frac{2}{2} & & = \frac{16}{2} \\ & = 1 & & = 8 \\ \\ V &= \Big\{ 1 ; 8 \Big\} & &\end{align} \\ -----------------\)
8. \(x(2x+79)=72(x+1) \\ \Leftrightarrow 2x^2+79x=72x+72 \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(-(14-10x)=-x^2-(18-8x) \\ \Leftrightarrow -14+10x=-x^2-18+8x \\ \Leftrightarrow x^2+2x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.4 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(x(x+20)=4(x-16) \\ \Leftrightarrow x^2+20x=4x-64 \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)
11. \(-(13-15x)=-x^2-(29-5x) \\ \Leftrightarrow -13+15x=-x^2-29+5x \\ \Leftrightarrow x^2+10x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.16 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.1} & & = \frac{-10+\sqrt36}{2.1} \\ & = \frac{-16}{2} & & = \frac{-4}{2} \\ & = -8 & & = -2 \\ \\ V &= \Big\{ -8 ; -2 \Big\} & &\end{align} \\ -----------------\)
12. \(-(2-x)=-x^2-(-54-2x) \\ \Leftrightarrow -2+x=-x^2+54+2x \\ \Leftrightarrow x^2-x-56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-56=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-56) & &\\ & = 1+224 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt225}{2.1} & & = \frac{-(-1)+\sqrt225}{2.1} \\ & = \frac{-14}{2} & & = \frac{16}{2} \\ & = -7 & & = 8 \\ \\ V &= \Big\{ -7 ; 8 \Big\} & &\end{align} \\ -----------------\)