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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(8-7x)=-x^2-(19-19x)\)
2. \(-3x=-\frac{2}{3}x^2-\frac{25}{6}\)
3. \(2x=-\frac{1}{5}x^2-\frac{21}{5}\)
4. \(-(11-36x)=-2x^2-(83-11x)\)
5. \(x(4x+4)=-9(x+1)\)
6. \(-(6+13x)=-x^2-(42-2x)\)
7. \(x(x+50)=48(x+1)\)
8. \(\frac{1}{5}x^2+\frac{3}{2}x+5=0\)
9. \(x(x+0)=2(x-8)\)
10. \(2x^2-(17x-16)=x(x-23)\)
11. \(-(8-19x)=-9x^2-(4-14x)\)
12. \((2x+5)(2x-2)-x(0x-34)=-35\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(8-7x)=-x^2-(19-19x) \\ \Leftrightarrow -8+7x=-x^2-19+19x \\ \Leftrightarrow x^2-12x+11=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+11=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.11 & &\\ & = 144-44 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt100}{2.1} & & = \frac{-(-12)+\sqrt100}{2.1} \\ & = \frac{2}{2} & & = \frac{22}{2} \\ & = 1 & & = 11 \\ \\ V &= \Big\{ 1 ; 11 \Big\} & &\end{align} \\ -----------------\)
2. \(-3x=-\frac{2}{3}x^2-\frac{25}{6} \\ \Leftrightarrow \frac{2}{3}x^2-3x+\frac{25}{6}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{2}{3}x^2-3x+\frac{25}{6}\right)=0 \color{red}{.6} \\ \Leftrightarrow 4x^2-18x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-18x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.4.25 & &\\ & = 324-400 & & \\ & = -76 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(2x=-\frac{1}{5}x^2-\frac{21}{5} \\ \Leftrightarrow \frac{1}{5}x^2+2x+\frac{21}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+2x+\frac{21}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+10x+21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.21 & &\\ & = 100-84 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt16}{2.1} & & = \frac{-10+\sqrt16}{2.1} \\ & = \frac{-14}{2} & & = \frac{-6}{2} \\ & = -7 & & = -3 \\ \\ V &= \Big\{ -7 ; -3 \Big\} & &\end{align} \\ -----------------\)
4. \(-(11-36x)=-2x^2-(83-11x) \\ \Leftrightarrow -11+36x=-2x^2-83+11x \\ \Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.2.72 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\ & = \frac{-32}{4} & & = \frac{-18}{4} \\ & = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(x(4x+4)=-9(x+1) \\ \Leftrightarrow 4x^2+4x=-9x-9 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
6. \(-(6+13x)=-x^2-(42-2x) \\ \Leftrightarrow -6-13x=-x^2-42+2x \\ \Leftrightarrow x^2-15x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.36 & &\\ & = 225-144 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt81}{2.1} & & = \frac{-(-15)+\sqrt81}{2.1} \\ & = \frac{6}{2} & & = \frac{24}{2} \\ & = 3 & & = 12 \\ \\ V &= \Big\{ 3 ; 12 \Big\} & &\end{align} \\ -----------------\)
7. \(x(x+50)=48(x+1) \\ \Leftrightarrow x^2+50x=48x+48 \\ \Leftrightarrow x^2+2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.1.(-48) & &\\ & = 4+192 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-2-\sqrt196}{2.1} & & = \frac{-2+\sqrt196}{2.1} \\ & = \frac{-16}{2} & & = \frac{12}{2} \\ & = -8 & & = 6 \\ \\ V &= \Big\{ -8 ; 6 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+\frac{3}{2}x+5=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{3}{2}x+5\right)=0 \color{red}{.10} \\ \Leftrightarrow 4x^2+30x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+30x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (30)^2-4.4.100 & &\\ & = 900-1600 & & \\ & = -700 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(x(x+0)=2(x-8) \\ \Leftrightarrow x^2+0x=2x-16 \\ \Leftrightarrow x^2-2x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.16 & &\\ & = 4-64 & & \\ & = -60 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(2x^2-(17x-16)=x(x-23) \\ \Leftrightarrow 2x^2-17x+16=x^2-23x \\ \Leftrightarrow x^2+6x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.16 & &\\ & = 36-64 & & \\ & = -28 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(-(8-19x)=-9x^2-(4-14x) \\ \Leftrightarrow -8+19x=-9x^2-4+14x \\ \Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.9.(-4) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\ & = \frac{-18}{18} & & = \frac{8}{18} \\ & = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
12. \((2x+5)(2x-2)-x(0x-34)=-35\\ \Leftrightarrow 4x^2-4x+10x-10 +0x^2+34x+35=0 \\ \Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.4} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)