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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((x+3)(x-4)-x(0x-21)=-16\)
2. \((3x-3)(3x-1)-x(8x-11)=15\)
3. \(\frac{1}{48}x^2-\frac{1}{3}x+\frac{4}{3}=0\)
4. \(8x^2-(17x+4)=7x(x-2)\)
5. \(\frac{1}{4}x^2-\frac{3}{4}x-\frac{5}{2}=0\)
6. \(-(11-39x)=-9x^2-(36-9x)\)
7. \(2x^2-(4x-2)=x(x-1)\)
8. \(x^2+\frac{17}{8}x+\frac{1}{4}=0\)
9. \(-(13-40x)=-9x^2-(157-8x)\)
10. \(x(18x+15)=8(x+1)\)
11. \(\frac{2}{3}x^2+\frac{5}{3}x+\frac{25}{24}=0\)
12. \((4x-3)(-2x-3)-x(-10x-8)=27\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((x+3)(x-4)-x(0x-21)=-16\\ \Leftrightarrow x^2-4x+3x-12 +0x^2+21x+16=0 \\ \Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\ & = \frac{-8}{2} & & = \frac{-2}{2} \\ & = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
2. \((3x-3)(3x-1)-x(8x-11)=15\\ \Leftrightarrow 9x^2-3x-9x+3 -8x^2+11x-15=0 \\ \Leftrightarrow x^2+11x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.(-12) & &\\ & = 121+48 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt169}{2.1} & & = \frac{-11+\sqrt169}{2.1} \\ & = \frac{-24}{2} & & = \frac{2}{2} \\ & = -12 & & = 1 \\ \\ V &= \Big\{ -12 ; 1 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{48}x^2-\frac{1}{3}x+\frac{4}{3}=0\\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2-\frac{1}{3}x+\frac{4}{3}\right)=0 \color{red}{.48} \\ \Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-16)}{2.1} & & \\ & = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
4. \(8x^2-(17x+4)=7x(x-2) \\ \Leftrightarrow 8x^2-17x-4=7x^2-14x \\ \Leftrightarrow x^2-3x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-4) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt25}{2.1} & & = \frac{-(-3)+\sqrt25}{2.1} \\ & = \frac{-2}{2} & & = \frac{8}{2} \\ & = -1 & & = 4 \\ \\ V &= \Big\{ -1 ; 4 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{4}x^2-\frac{3}{4}x-\frac{5}{2}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-\frac{3}{4}x-\frac{5}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow x^2-3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.(-10) & &\\ & = 9+40 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt49}{2.1} & & = \frac{-(-3)+\sqrt49}{2.1} \\ & = \frac{-4}{2} & & = \frac{10}{2} \\ & = -2 & & = 5 \\ \\ V &= \Big\{ -2 ; 5 \Big\} & &\end{align} \\ -----------------\)
6. \(-(11-39x)=-9x^2-(36-9x) \\ \Leftrightarrow -11+39x=-9x^2-36+9x \\ \Leftrightarrow 9x^2+30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-30}{2.9} & & \\ & = -\frac{5}{3} & & \\V &= \Big\{ -\frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
7. \(2x^2-(4x-2)=x(x-1) \\ \Leftrightarrow 2x^2-4x+2=x^2-x \\ \Leftrightarrow x^2-3x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-3)^2-4.1.2 & &\\ & = 9-8 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-3)-\sqrt1}{2.1} & & = \frac{-(-3)+\sqrt1}{2.1} \\ & = \frac{2}{2} & & = \frac{4}{2} \\ & = 1 & & = 2 \\ \\ V &= \Big\{ 1 ; 2 \Big\} & &\end{align} \\ -----------------\)
8. \(x^2+\frac{17}{8}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{17}{8}x+\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(-(13-40x)=-9x^2-(157-8x) \\ \Leftrightarrow -13+40x=-9x^2-157+8x \\ \Leftrightarrow 9x^2+32x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+32x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (32)^2-4.9.144 & &\\ & = 1024-5184 & & \\ & = -4160 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
10. \(x(18x+15)=8(x+1) \\ \Leftrightarrow 18x^2+15x=8x+8 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{2}{3}x^2+\frac{5}{3}x+\frac{25}{24}=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2+\frac{5}{3}x+\frac{25}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2+40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.16} & & \\ & = -\frac{5}{4} & & \\V &= \Big\{ -\frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
12. \((4x-3)(-2x-3)-x(-10x-8)=27\\ \Leftrightarrow -8x^2-12x+6x+9 +10x^2+8x-27=0 \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)