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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((2x+5)(3x-4)-x(4x-35)=52\)
2. \(x(x-10)=3(x-10)\)
3. \(\frac{5}{4}x=-\frac{1}{2}x^2+\frac{9}{2}\)
4. \(-(13+16x)=-x^2-(157-8x)\)
5. \((-4x+4)(-2x+2)-x(4x-8)=-56\)
6. \(x(12x+10)=-3(x+1)\)
7. \(\frac{16}{45}x^2-\frac{8}{5}x+\frac{9}{5}=0\)
8. \(-(8-37x)=-8x^2-(10-20x)\)
9. \(2x^2-(8x-14)=x(x-17)\)
10. \(-(8-8x)=-x^2-(13-14x)\)
11. \(75x^2-(20x-2)=3x(x-15)\)
12. \(\frac{1}{3}x^2+4x+\frac{35}{3}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((2x+5)(3x-4)-x(4x-35)=52\\ \Leftrightarrow 6x^2-8x+15x-20 -4x^2+35x-52=0 \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(x(x-10)=3(x-10) \\ \Leftrightarrow x^2-10x=3x-30 \\ \Leftrightarrow x^2-13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.30 & &\\ & = 169-120 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt49}{2.1} & & = \frac{-(-13)+\sqrt49}{2.1} \\ & = \frac{6}{2} & & = \frac{20}{2} \\ & = 3 & & = 10 \\ \\ V &= \Big\{ 3 ; 10 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{5}{4}x=-\frac{1}{2}x^2+\frac{9}{2} \\ \Leftrightarrow \frac{1}{2}x^2+\frac{5}{4}x-\frac{9}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{5}{4}x-\frac{9}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \(-(13+16x)=-x^2-(157-8x) \\ \Leftrightarrow -13-16x=-x^2-157+8x \\ \Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-24)^2-4.1.144 & &\\ & = 576-576 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-24)}{2.1} & & \\ & = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
5. \((-4x+4)(-2x+2)-x(4x-8)=-56\\ \Leftrightarrow 8x^2-8x-8x+8 -4x^2+8x+56=0 \\ \Leftrightarrow 4x^2+8x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+8x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.4.64 & &\\ & = 64-1024 & & \\ & = -960 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(x(12x+10)=-3(x+1) \\ \Leftrightarrow 12x^2+10x=-3x-3 \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{16}{45}x^2-\frac{8}{5}x+\frac{9}{5}=0\\ \Leftrightarrow \color{red}{45.} \left(\frac{16}{45}x^2-\frac{8}{5}x+\frac{9}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow 16x^2-72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-72)}{2.16} & & \\ & = \frac{9}{4} & & \\V &= \Big\{ \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(-(8-37x)=-8x^2-(10-20x) \\ \Leftrightarrow -8+37x=-8x^2-10+20x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2-(8x-14)=x(x-17) \\ \Leftrightarrow 2x^2-8x+14=x^2-17x \\ \Leftrightarrow x^2+9x+14=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x+14=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.14 & &\\ & = 81-56 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt25}{2.1} & & = \frac{-9+\sqrt25}{2.1} \\ & = \frac{-14}{2} & & = \frac{-4}{2} \\ & = -7 & & = -2 \\ \\ V &= \Big\{ -7 ; -2 \Big\} & &\end{align} \\ -----------------\)
10. \(-(8-8x)=-x^2-(13-14x) \\ \Leftrightarrow -8+8x=-x^2-13+14x \\ \Leftrightarrow x^2-6x+5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.5 & &\\ & = 36-20 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt16}{2.1} & & = \frac{-(-6)+\sqrt16}{2.1} \\ & = \frac{2}{2} & & = \frac{10}{2} \\ & = 1 & & = 5 \\ \\ V &= \Big\{ 1 ; 5 \Big\} & &\end{align} \\ -----------------\)
11. \(75x^2-(20x-2)=3x(x-15) \\ \Leftrightarrow 75x^2-20x+2=3x^2-45x \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{1}{3}x^2+4x+\frac{35}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+4x+\frac{35}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+12x+35=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+35=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.35 & &\\ & = 144-140 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt4}{2.1} & & = \frac{-12+\sqrt4}{2.1} \\ & = \frac{-14}{2} & & = \frac{-10}{2} \\ & = -7 & & = -5 \\ \\ V &= \Big\{ -7 ; -5 \Big\} & &\end{align} \\ -----------------\)