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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(4-23x)=-16x^2-(5-15x)\)
2. \((3x-3)(4x+1)-x(-60x-25)=-5\)
3. \((-x+3)(-3x-1)-x(x-15)=-21\)
4. \(17x^2-(18x-36)=8x(x-4)\)
5. \((2x+5)(-4x+3)-x(-80x+14)=17\)
6. \(\frac{1}{21}x^2+\frac{2}{3}x+\frac{7}{3}=0\)
7. \(x(16x-84)=-36(x+1)\)
8. \(-(4-30x)=-4x^2-(148-14x)\)
9. \(3x^2-(17x+10)=2x(x-13)\)
10. \(-(14-5x)=-x^2-(23-11x)\)
11. \((-3x-4)(2x-2)-x(-15x+54)=-136\)
12. \(\frac{1}{6}x^2+x+\frac{3}{2}=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(4-23x)=-16x^2-(5-15x) \\ \Leftrightarrow -4+23x=-16x^2-5+15x \\ \Leftrightarrow 16x^2+8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.16} & & \\ & = -\frac{1}{4} & & \\V &= \Big\{ -\frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \((3x-3)(4x+1)-x(-60x-25)=-5\\ \Leftrightarrow 12x^2+3x-12x-3 +60x^2+25x+5=0 \\ \Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.72.2 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\ & = \frac{-32}{144} & & = \frac{-18}{144} \\ & = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
3. \((-x+3)(-3x-1)-x(x-15)=-21\\ \Leftrightarrow 3x^2+x-9x-3 -x^2+15x+21=0 \\ \Leftrightarrow 2x^2+13x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+13x+18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.2.18 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.2} & & = \frac{-13+\sqrt25}{2.2} \\ & = \frac{-18}{4} & & = \frac{-8}{4} \\ & = \frac{-9}{2} & & = -2 \\ \\ V &= \Big\{ \frac{-9}{2} ; -2 \Big\} & &\end{align} \\ -----------------\)
4. \(17x^2-(18x-36)=8x(x-4) \\ \Leftrightarrow 17x^2-18x+36=8x^2-32x \\ \Leftrightarrow 9x^2+14x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+14x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.9.36 & &\\ & = 196-1296 & & \\ & = -1100 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \((2x+5)(-4x+3)-x(-80x+14)=17\\ \Leftrightarrow -8x^2+6x-20x+15 +80x^2-14x-17=0 \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{1}{21}x^2+\frac{2}{3}x+\frac{7}{3}=0\\ \Leftrightarrow \color{red}{21.} \left(\frac{1}{21}x^2+\frac{2}{3}x+\frac{7}{3}\right)=0 \color{red}{.21} \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
7. \(x(16x-84)=-36(x+1) \\ \Leftrightarrow 16x^2-84x=-36x-36 \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(-(4-30x)=-4x^2-(148-14x) \\ \Leftrightarrow -4+30x=-4x^2-148+14x \\ \Leftrightarrow 4x^2+16x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.144 & &\\ & = 256-2304 & & \\ & = -2048 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(3x^2-(17x+10)=2x(x-13) \\ \Leftrightarrow 3x^2-17x-10=2x^2-26x \\ \Leftrightarrow x^2+9x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+9x-10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (9)^2-4.1.(-10) & &\\ & = 81+40 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-9-\sqrt121}{2.1} & & = \frac{-9+\sqrt121}{2.1} \\ & = \frac{-20}{2} & & = \frac{2}{2} \\ & = -10 & & = 1 \\ \\ V &= \Big\{ -10 ; 1 \Big\} & &\end{align} \\ -----------------\)
10. \(-(14-5x)=-x^2-(23-11x) \\ \Leftrightarrow -14+5x=-x^2-23+11x \\ \Leftrightarrow x^2-6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.1} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
11. \((-3x-4)(2x-2)-x(-15x+54)=-136\\ \Leftrightarrow -6x^2+6x-8x+8 +15x^2-54x+136=0 \\ \Leftrightarrow 9x^2-40x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-40x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-40)^2-4.9.144 & &\\ & = 1600-5184 & & \\ & = -3584 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(\frac{1}{6}x^2+x+\frac{3}{2}=0\\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+x+\frac{3}{2}\right)=0 \color{red}{.6} \\ \Leftrightarrow 9x^2+54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-54}{2.9} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)