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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((x+3)(5x-1)-x(x-9)=-4\)
2. \(-(2-16x)=-4x^2-(11-20x)\)
3. \(\frac{1}{10}x^2-\frac{7}{5}x+4=0\)
4. \(-(13+14x)=-4x^2-(29-2x)\)
5. \((-4x-5)(-x+5)-x(3x-61)=-73\)
6. \(x(x+97)=99(x+1)\)
7. \(2x^2-(19x+48)=x(x-27)\)
8. \(\frac{4}{3}x=-\frac{4}{27}x^2-3\)
9. \(x(16x+33)=5(x-5)\)
10. \(x(4x+4)=-9(x+1)\)
11. \(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}=0\)
12. \(-(15+66x)=-16x^2-(115-14x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((x+3)(5x-1)-x(x-9)=-4\\ \Leftrightarrow 5x^2-x+15x-3 -x^2+9x+4=0 \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(-(2-16x)=-4x^2-(11-20x) \\ \Leftrightarrow -2+16x=-4x^2-11+20x \\ \Leftrightarrow 4x^2-4x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.9 & &\\ & = 16-144 & & \\ & = -128 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(\frac{1}{10}x^2-\frac{7}{5}x+4=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{10}x^2-\frac{7}{5}x+4\right)=0 \color{red}{.10} \\ \Leftrightarrow x^2-14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
4. \(-(13+14x)=-4x^2-(29-2x) \\ \Leftrightarrow -13-14x=-4x^2-29+2x \\ \Leftrightarrow 4x^2-16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.4} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
5. \((-4x-5)(-x+5)-x(3x-61)=-73\\ \Leftrightarrow 4x^2-20x+5x-25 -3x^2+61x+73=0 \\ \Leftrightarrow x^2+16x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.48 & &\\ & = 256-192 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt64}{2.1} & & = \frac{-16+\sqrt64}{2.1} \\ & = \frac{-24}{2} & & = \frac{-8}{2} \\ & = -12 & & = -4 \\ \\ V &= \Big\{ -12 ; -4 \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+97)=99(x+1) \\ \Leftrightarrow x^2+97x=99x+99 \\ \Leftrightarrow x^2-2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-99=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.(-99) & &\\ & = 4+396 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-2)-\sqrt400}{2.1} & & = \frac{-(-2)+\sqrt400}{2.1} \\ & = \frac{-18}{2} & & = \frac{22}{2} \\ & = -9 & & = 11 \\ \\ V &= \Big\{ -9 ; 11 \Big\} & &\end{align} \\ -----------------\)
7. \(2x^2-(19x+48)=x(x-27) \\ \Leftrightarrow 2x^2-19x-48=x^2-27x \\ \Leftrightarrow x^2+8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+8x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.1.(-48) & &\\ & = 64+192 & & \\ & = 256 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-8-\sqrt256}{2.1} & & = \frac{-8+\sqrt256}{2.1} \\ & = \frac{-24}{2} & & = \frac{8}{2} \\ & = -12 & & = 4 \\ \\ V &= \Big\{ -12 ; 4 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{4}{3}x=-\frac{4}{27}x^2-3 \\ \Leftrightarrow \frac{4}{27}x^2+\frac{4}{3}x+3=0 \\ \Leftrightarrow \color{red}{27.} \left(\frac{4}{27}x^2+\frac{4}{3}x+3\right)=0 \color{red}{.27} \\ \Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.4.81 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.4} & & \\ & = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
9. \(x(16x+33)=5(x-5) \\ \Leftrightarrow 16x^2+33x=5x-25 \\ \Leftrightarrow 16x^2+28x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+28x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.16.25 & &\\ & = 784-1600 & & \\ & = -816 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(x(4x+4)=-9(x+1) \\ \Leftrightarrow 4x^2+4x=-9x-9 \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{32}x^2-\frac{1}{4}x+\frac{1}{2}\right)=0 \color{red}{.32} \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
12. \(-(15+66x)=-16x^2-(115-14x) \\ \Leftrightarrow -15-66x=-16x^2-115+14x \\ \Leftrightarrow 16x^2-80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-80)}{2.16} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)