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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{3}x^2-\frac{2}{3}x-33=0\)
2. \(\frac{9}{20}x^2-\frac{3}{2}x+\frac{5}{4}=0\)
3. \((-2x-5)(-3x+5)-x(-10x+45)=-125\)
4. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2}\)
5. \(-(11-8x)=-x^2-(6-12x)\)
6. \(-(15-9x)=-x^2-(-18-17x)\)
7. \(x(9x+14)=8(x-2)\)
8. \(\frac{1}{3}x^2+\frac{1}{3}x-24=0\)
9. \(\frac{1}{4}x=-\frac{1}{4}x^2+\frac{45}{2}\)
10. \(11x^2-(16x-25)=2x(x-18)\)
11. \(-\frac{21}{4}x=-\frac{1}{4}x^2-\frac{55}{2}\)
12. \(2x^2-(17x-36)=x(x-32)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{3}x^2-\frac{2}{3}x-33=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{2}{3}x-33\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt400}{2.1} & & = \frac{-(-2)+\sqrt400}{2.1} \\ & = \frac{-18}{2} & & = \frac{22}{2} \\ & = -9 & & = 11 \\ \\ V &= \Big\{ -9 ; 11 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{9}{20}x^2-\frac{3}{2}x+\frac{5}{4}=0\\ \Leftrightarrow \color{red}{20.} \left(\frac{9}{20}x^2-\frac{3}{2}x+\frac{5}{4}\right)=0 \color{red}{.20} \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
3. \((-2x-5)(-3x+5)-x(-10x+45)=-125\\ \Leftrightarrow 6x^2-10x+15x-25 +10x^2-45x+125=0 \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2} \\ \Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 4x^2-16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.4} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
5. \(-(11-8x)=-x^2-(6-12x) \\ \Leftrightarrow -11+8x=-x^2-6+12x \\ \Leftrightarrow x^2-4x-5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-5) & &\\ & = 16+20 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt36}{2.1} & & = \frac{-(-4)+\sqrt36}{2.1} \\ & = \frac{-2}{2} & & = \frac{10}{2} \\ & = -1 & & = 5 \\ \\ V &= \Big\{ -1 ; 5 \Big\} & &\end{align} \\ -----------------\)
6. \(-(15-9x)=-x^2-(-18-17x) \\ \Leftrightarrow -15+9x=-x^2+18+17x \\ \Leftrightarrow x^2-8x-33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-33) & &\\ & = 64+132 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt196}{2.1} & & = \frac{-(-8)+\sqrt196}{2.1} \\ & = \frac{-6}{2} & & = \frac{22}{2} \\ & = -3 & & = 11 \\ \\ V &= \Big\{ -3 ; 11 \Big\} & &\end{align} \\ -----------------\)
7. \(x(9x+14)=8(x-2) \\ \Leftrightarrow 9x^2+14x=8x-16 \\ \Leftrightarrow 9x^2+6x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+6x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.9.16 & &\\ & = 36-576 & & \\ & = -540 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(\frac{1}{3}x^2+\frac{1}{3}x-24=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{1}{3}x-24\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt289}{2.1} & & = \frac{-1+\sqrt289}{2.1} \\ & = \frac{-18}{2} & & = \frac{16}{2} \\ & = -9 & & = 8 \\ \\ V &= \Big\{ -9 ; 8 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{4}x=-\frac{1}{4}x^2+\frac{45}{2} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{1}{4}x-\frac{45}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{1}{4}x-\frac{45}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2+x-90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-90=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (1)^2-4.1.(-90) & &\\ & = 1+360 & & \\ & = 361 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-1-\sqrt361}{2.1} & & = \frac{-1+\sqrt361}{2.1} \\ & = \frac{-20}{2} & & = \frac{18}{2} \\ & = -10 & & = 9 \\ \\ V &= \Big\{ -10 ; 9 \Big\} & &\end{align} \\ -----------------\)
10. \(11x^2-(16x-25)=2x(x-18) \\ \Leftrightarrow 11x^2-16x+25=2x^2-36x \\ \Leftrightarrow 9x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.9.25 & &\\ & = 400-900 & & \\ & = -500 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(-\frac{21}{4}x=-\frac{1}{4}x^2-\frac{55}{2} \\ \Leftrightarrow \frac{1}{4}x^2-\frac{21}{4}x+\frac{55}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{21}{4}x+\frac{55}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+110=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.110 & &\\ & = 441-440 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt1}{2.1} & & = \frac{-(-21)+\sqrt1}{2.1} \\ & = \frac{20}{2} & & = \frac{22}{2} \\ & = 10 & & = 11 \\ \\ V &= \Big\{ 10 ; 11 \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2-(17x-36)=x(x-32) \\ \Leftrightarrow 2x^2-17x+36=x^2-32x \\ \Leftrightarrow x^2+15x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.36 & &\\ & = 225-144 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt81}{2.1} & & = \frac{-15+\sqrt81}{2.1} \\ & = \frac{-24}{2} & & = \frac{-6}{2} \\ & = -12 & & = -3 \\ \\ V &= \Big\{ -12 ; -3 \Big\} & &\end{align} \\ -----------------\)