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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2-(2x-10)=x(x-13)\)
2. \(\frac{1}{3}x^2+\frac{25}{9}x+\frac{16}{3}=0\)
3. \(x(x+7)=10(x+1)\)
4. \(-(9-8x)=-2x^2-(11-3x)\)
5. \(-\frac{2}{3}x=-\frac{1}{3}x^2+1\)
6. \(x(x+61)=66(x+1)\)
7. \((-5x+3)(-5x-1)-x(24x+14)=-39\)
8. \(x=-\frac{1}{6}x^2-\frac{3}{2}\)
9. \(x(16x-27)=7(x-7)\)
10. \(\frac{2}{9}x^2+2x+\frac{9}{2}=0\)
11. \(x(9x-33)=3(x-12)\)
12. \(x(x-19)=-33(x+1)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(2x^2-(2x-10)=x(x-13) \\ \Leftrightarrow 2x^2-2x+10=x^2-13x \\ \Leftrightarrow x^2+11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt81}{2.1} & & = \frac{-11+\sqrt81}{2.1} \\ & = \frac{-20}{2} & & = \frac{-2}{2} \\ & = -10 & & = -1 \\ \\ V &= \Big\{ -10 ; -1 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{3}x^2+\frac{25}{9}x+\frac{16}{3}=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{3}x^2+\frac{25}{9}x+\frac{16}{3}\right)=0 \color{red}{.9} \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
3. \(x(x+7)=10(x+1) \\ \Leftrightarrow x^2+7x=10x+10 \\ \Leftrightarrow x^2-3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt49}{2.1} & & = \frac{-(-3)+\sqrt49}{2.1} \\ & = \frac{-4}{2} & & = \frac{10}{2} \\ & = -2 & & = 5 \\ \\ V &= \Big\{ -2 ; 5 \Big\} & &\end{align} \\ -----------------\)
4. \(-(9-8x)=-2x^2-(11-3x) \\ \Leftrightarrow -9+8x=-2x^2-11+3x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(-\frac{2}{3}x=-\frac{1}{3}x^2+1 \\ \Leftrightarrow \frac{1}{3}x^2-\frac{2}{3}x-1=0 \\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{2}{3}x-1\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2-2x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-3) & &\\ & = 4+12 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt16}{2.1} & & = \frac{-(-2)+\sqrt16}{2.1} \\ & = \frac{-2}{2} & & = \frac{6}{2} \\ & = -1 & & = 3 \\ \\ V &= \Big\{ -1 ; 3 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+61)=66(x+1) \\ \Leftrightarrow x^2+61x=66x+66 \\ \Leftrightarrow x^2-5x-66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.(-66) & &\\ & = 25+264 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt289}{2.1} & & = \frac{-(-5)+\sqrt289}{2.1} \\ & = \frac{-12}{2} & & = \frac{22}{2} \\ & = -6 & & = 11 \\ \\ V &= \Big\{ -6 ; 11 \Big\} & &\end{align} \\ -----------------\)
7. \((-5x+3)(-5x-1)-x(24x+14)=-39\\ \Leftrightarrow 25x^2+5x-15x-3 -24x^2-14x+39=0 \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
8. \(x=-\frac{1}{6}x^2-\frac{3}{2} \\ \Leftrightarrow \frac{1}{6}x^2+x+\frac{3}{2}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+x+\frac{3}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
9. \(x(16x-27)=7(x-7) \\ \Leftrightarrow 16x^2-27x=7x-49 \\ \Leftrightarrow 16x^2-34x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-34x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-34)^2-4.16.49 & &\\ & = 1156-3136 & & \\ & = -1980 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(\frac{2}{9}x^2+2x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{18.} \left(\frac{2}{9}x^2+2x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(x(9x-33)=3(x-12) \\ \Leftrightarrow 9x^2-33x=3x-36 \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
12. \(x(x-19)=-33(x+1) \\ \Leftrightarrow x^2-19x=-33x-33 \\ \Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\ & = \frac{-22}{2} & & = \frac{-6}{2} \\ & = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)