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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(3x^2-(20x-8)=x(x-37)\)
2. \(\frac{1}{4}x^2-\frac{1}{2}x-30=0\)
3. \(-(5-15x)=-2x^2-(7-10x)\)
4. \(-(14+54x)=-9x^2-(158-18x)\)
5. \(\frac{7}{4}x=-18x^2+\frac{1}{2}\)
6. \(x(x-155)=-132(x+1)\)
7. \(8x^2-(20x-9)=4x(x-2)\)
8. \(x(x-13)=3(x-21)\)
9. \(5x^2-(20x-81)=4x(x-3)\)
10. \(\frac{1}{22}x^2+x+\frac{11}{2}=0\)
11. \(x(4x+52)=4(x-36)\)
12. \(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(3x^2-(20x-8)=x(x-37) \\ \Leftrightarrow 3x^2-20x+8=x^2-37x \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{4}x^2-\frac{1}{2}x-30=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{1}{2}x-30\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-2x-120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-120=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-120) & &\\ & = 4+480 & & \\ & = 484 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt484}{2.1} & & = \frac{-(-2)+\sqrt484}{2.1} \\ & = \frac{-20}{2} & & = \frac{24}{2} \\ & = -10 & & = 12 \\ \\ V &= \Big\{ -10 ; 12 \Big\} & &\end{align} \\ -----------------\)
3. \(-(5-15x)=-2x^2-(7-10x) \\ \Leftrightarrow -5+15x=-2x^2-7+10x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(-(14+54x)=-9x^2-(158-18x) \\ \Leftrightarrow -14-54x=-9x^2-158+18x \\ \Leftrightarrow 9x^2-72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-72x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.9.144 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.9} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{7}{4}x=-18x^2+\frac{1}{2} \\ \Leftrightarrow 18x^2+\frac{7}{4}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(18x^2+\frac{7}{4}x-\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(x(x-155)=-132(x+1) \\ \Leftrightarrow x^2-155x=-132x-132 \\ \Leftrightarrow x^2-23x+132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-23x+132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-23)^2-4.1.132 & &\\ & = 529-528 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-23)-\sqrt1}{2.1} & & = \frac{-(-23)+\sqrt1}{2.1} \\ & = \frac{22}{2} & & = \frac{24}{2} \\ & = 11 & & = 12 \\ \\ V &= \Big\{ 11 ; 12 \Big\} & &\end{align} \\ -----------------\)
7. \(8x^2-(20x-9)=4x(x-2) \\ \Leftrightarrow 8x^2-20x+9=4x^2-8x \\ \Leftrightarrow 4x^2-12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.4} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x(x-13)=3(x-21) \\ \Leftrightarrow x^2-13x=3x-63 \\ \Leftrightarrow x^2-16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+63=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.63 & &\\ & = 256-252 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt4}{2.1} & & = \frac{-(-16)+\sqrt4}{2.1} \\ & = \frac{14}{2} & & = \frac{18}{2} \\ & = 7 & & = 9 \\ \\ V &= \Big\{ 7 ; 9 \Big\} & &\end{align} \\ -----------------\)
9. \(5x^2-(20x-81)=4x(x-3) \\ \Leftrightarrow 5x^2-20x+81=4x^2-12x \\ \Leftrightarrow x^2-8x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.81 & &\\ & = 64-324 & & \\ & = -260 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(\frac{1}{22}x^2+x+\frac{11}{2}=0\\ \Leftrightarrow \color{red}{22.} \left(\frac{1}{22}x^2+x+\frac{11}{2}\right)=0 \color{red}{.22} \\ \Leftrightarrow x^2+22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-22}{2.1} & & \\ & = -11 & & \\V &= \Big\{ -11 \Big\} & &\end{align} \\ -----------------\)
11. \(x(4x+52)=4(x-36) \\ \Leftrightarrow 4x^2+52x=4x-144 \\ \Leftrightarrow 4x^2+48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.4} & & \\ & = -6 & & \\V &= \Big\{ -6 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2-\frac{1}{4}x+\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 4x^2-16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.4} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)