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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \((-5x-3)(5x-1)-x(-28x+1)=51\)
2. \(3x^2-(5x-40)=2x(x-9)\)
3. \((5x-4)(x+3)-x(x-13)=-28\)
4. \(-(12-27x)=-16x^2-(13-10x)\)
5. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\)
6. \(x(x+19)=8(x-3)\)
7. \(-(6-24x)=-x^2-(66-8x)\)
8. \(x(9x-51)=3(x-27)\)
9. \(x(18x+11)=-2(x+1)\)
10. \(-(7-9x)=-2x^2-(-65-2x)\)
11. \(21x^2-(9x-2)=13x(x-2)\)
12. \(x(x-48)=-64(x+1)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \((-5x-3)(5x-1)-x(-28x+1)=51\\ \Leftrightarrow -25x^2+5x-15x+3 +28x^2-x-51=0 \\ \Leftrightarrow 3x^2+7x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+7x-48=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.3.(-48) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.3} & & = \frac{-7+\sqrt625}{2.3} \\ & = \frac{-32}{6} & & = \frac{18}{6} \\ & = \frac{-16}{3} & & = 3 \\ \\ V &= \Big\{ \frac{-16}{3} ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(3x^2-(5x-40)=2x(x-9) \\ \Leftrightarrow 3x^2-5x+40=2x^2-18x \\ \Leftrightarrow x^2+13x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.1.40 & &\\ & = 169-160 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt9}{2.1} & & = \frac{-13+\sqrt9}{2.1} \\ & = \frac{-16}{2} & & = \frac{-10}{2} \\ & = -8 & & = -5 \\ \\ V &= \Big\{ -8 ; -5 \Big\} & &\end{align} \\ -----------------\)
3. \((5x-4)(x+3)-x(x-13)=-28\\ \Leftrightarrow 5x^2+15x-4x-12 -x^2+13x+28=0 \\ \Leftrightarrow 4x^2+16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.4.16 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.4} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
4. \(-(12-27x)=-16x^2-(13-10x) \\ \Leftrightarrow -12+27x=-16x^2-13+10x \\ \Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.16.1 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\ & = \frac{-32}{32} & & = \frac{-2}{32} \\ & = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{2}x^2+\frac{5}{12}x-\frac{1}{2}\right)=0 \color{red}{.12} \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
6. \(x(x+19)=8(x-3) \\ \Leftrightarrow x^2+19x=8x-24 \\ \Leftrightarrow x^2+11x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (11)^2-4.1.24 & &\\ & = 121-96 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-11-\sqrt25}{2.1} & & = \frac{-11+\sqrt25}{2.1} \\ & = \frac{-16}{2} & & = \frac{-6}{2} \\ & = -8 & & = -3 \\ \\ V &= \Big\{ -8 ; -3 \Big\} & &\end{align} \\ -----------------\)
7. \(-(6-24x)=-x^2-(66-8x) \\ \Leftrightarrow -6+24x=-x^2-66+8x \\ \Leftrightarrow x^2+16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-16-\sqrt16}{2.1} & & = \frac{-16+\sqrt16}{2.1} \\ & = \frac{-20}{2} & & = \frac{-12}{2} \\ & = -10 & & = -6 \\ \\ V &= \Big\{ -10 ; -6 \Big\} & &\end{align} \\ -----------------\)
8. \(x(9x-51)=3(x-27) \\ \Leftrightarrow 9x^2-51x=3x-81 \\ \Leftrightarrow 9x^2-54x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-54x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-54)^2-4.9.81 & &\\ & = 2916-2916 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-54)}{2.9} & & \\ & = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
9. \(x(18x+11)=-2(x+1) \\ \Leftrightarrow 18x^2+11x=-2x-2 \\ \Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.18.2 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\ & = \frac{-18}{36} & & = \frac{-8}{36} \\ & = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
10. \(-(7-9x)=-2x^2-(-65-2x) \\ \Leftrightarrow -7+9x=-2x^2+65+2x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(21x^2-(9x-2)=13x(x-2) \\ \Leftrightarrow 21x^2-9x+2=13x^2-26x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
12. \(x(x-48)=-64(x+1) \\ \Leftrightarrow x^2-48x=-64x-64 \\ \Leftrightarrow x^2+16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.64 & &\\ & = 256-256 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-16}{2.1} & & \\ & = -8 & & \\V &= \Big\{ -8 \Big\} & &\end{align} \\ -----------------\)