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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-x-2)(-2x+2)-x(x+16)=-125\)
2. \((x-2)(-3x+2)-x(-9x-9)=20\)
3. \(-\frac{1}{2}x=-\frac{1}{2}x^2-\frac{25}{32}\)
4. \(x(x+8)=3(x-2)\)
5. \(13x^2-(9x-3)=x(x-22)\)
6. \(x(16x+9)=-(x+1)\)
7. \(-(11-22x)=-16x^2-(12-5x)\)
8. \(x(12x+19)=12(x+1)\)
9. \(5x^2-(10x-49)=x(x+18)\)
10. \((-2x+3)(-2x-3)-x(3x-23)=-109\)
11. \((x-4)(-5x-3)-x(-8x-16)=-36\)
12. \((-5x+5)(2x+5)-x(-11x-10)=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-x-2)(-2x+2)-x(x+16)=-125\\ \Leftrightarrow 2x^2-2x+4x-4 -x^2-16x+125=0 \\ \Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.1.121 & &\\ & = 484-484 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-22)}{2.1} & & \\ & = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
2. \((x-2)(-3x+2)-x(-9x-9)=20\\ \Leftrightarrow -3x^2+2x+6x-4 +9x^2+9x-20=0 \\ \Leftrightarrow 6x^2+7x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+7x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.6.(-24) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.6} & & = \frac{-7+\sqrt625}{2.6} \\ & = \frac{-32}{12} & & = \frac{18}{12} \\ & = \frac{-8}{3} & & = \frac{3}{2} \\ \\ V &= \Big\{ \frac{-8}{3} ; \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(-\frac{1}{2}x=-\frac{1}{2}x^2-\frac{25}{32} \\ \Leftrightarrow \frac{1}{2}x^2-\frac{1}{2}x+\frac{25}{32}=0 \\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{2}x^2-\frac{1}{2}x+\frac{25}{32}\right)=0 \color{red}{.32} \\ \Leftrightarrow 16x^2-16x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.25 & &\\ & = 256-1600 & & \\ & = -1344 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(x(x+8)=3(x-2) \\ \Leftrightarrow x^2+8x=3x-6 \\ \Leftrightarrow x^2+5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt1}{2.1} & & = \frac{-5+\sqrt1}{2.1} \\ & = \frac{-6}{2} & & = \frac{-4}{2} \\ & = -3 & & = -2 \\ \\ V &= \Big\{ -3 ; -2 \Big\} & &\end{align} \\ -----------------\)
5. \(13x^2-(9x-3)=x(x-22) \\ \Leftrightarrow 13x^2-9x+3=x^2-22x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(x(16x+9)=-(x+1) \\ \Leftrightarrow 16x^2+9x=-x-1 \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(-(11-22x)=-16x^2-(12-5x) \\ \Leftrightarrow -11+22x=-16x^2-12+5x \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
8. \(x(12x+19)=12(x+1) \\ \Leftrightarrow 12x^2+19x=12x+12 \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
9. \(5x^2-(10x-49)=x(x+18) \\ \Leftrightarrow 5x^2-10x+49=x^2+18x \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
10. \((-2x+3)(-2x-3)-x(3x-23)=-109\\ \Leftrightarrow 4x^2+6x-6x-9 -3x^2+23x+109=0 \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
11. \((x-4)(-5x-3)-x(-8x-16)=-36\\ \Leftrightarrow -5x^2-3x+20x+12 +8x^2+16x+36=0 \\ \Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.3.48 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\ & = \frac{-32}{6} & & = \frac{-18}{6} \\ & = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
12. \((-5x+5)(2x+5)-x(-11x-10)=0\\ \Leftrightarrow -10x^2-25x+10x+25 +11x^2+10x+0=0 \\ \Leftrightarrow x^2+10x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+10x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.1.25 & &\\ & = 100-100 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-10}{2.1} & & \\ & = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)