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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{4}{3}x^2+x-\frac{1}{3}=0\)
2. \(-(2-3x)=-x^2-(-43-7x)\)
3. \(-(6-13x)=-x^2-(10-9x)\)
4. \((-3x+3)(2x+2)-x(-7x+10)=30\)
5. \(\frac{8}{5}x=-\frac{16}{45}x^2-\frac{9}{5}\)
6. \(4x^2-(9x-9)=3x(x-5)\)
7. \(\frac{15}{4}x=-2x^2+\frac{1}{2}\)
8. \(2x^2-(9x-16)=x(x-5)\)
9. \(x(2x+21)=4(x-2)\)
10. \(x(9x+78)=8(x-18)\)
11. \(5x^2-(18x-1)=x(x-20)\)
12. \(8x^2-(9x-6)=7x(x-2)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{4}{3}x^2+x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{4}{3}x^2+x-\frac{1}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \(-(2-3x)=-x^2-(-43-7x) \\ \Leftrightarrow -2+3x=-x^2+43+7x \\ \Leftrightarrow x^2-4x-45=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-45=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-45) & &\\ & = 16+180 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt196}{2.1} & & = \frac{-(-4)+\sqrt196}{2.1} \\ & = \frac{-10}{2} & & = \frac{18}{2} \\ & = -5 & & = 9 \\ \\ V &= \Big\{ -5 ; 9 \Big\} & &\end{align} \\ -----------------\)
3. \(-(6-13x)=-x^2-(10-9x) \\ \Leftrightarrow -6+13x=-x^2-10+9x \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
4. \((-3x+3)(2x+2)-x(-7x+10)=30\\ \Leftrightarrow -6x^2-6x+6x+6 +7x^2-10x-30=0 \\ \Leftrightarrow x^2-10x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-24=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.(-24) & &\\ & = 100+96 & & \\ & = 196 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-10)-\sqrt196}{2.1} & & = \frac{-(-10)+\sqrt196}{2.1} \\ & = \frac{-4}{2} & & = \frac{24}{2} \\ & = -2 & & = 12 \\ \\ V &= \Big\{ -2 ; 12 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{8}{5}x=-\frac{16}{45}x^2-\frac{9}{5} \\ \Leftrightarrow \frac{16}{45}x^2+\frac{8}{5}x+\frac{9}{5}=0 \\ \Leftrightarrow \color{red}{45.} \left(\frac{16}{45}x^2+\frac{8}{5}x+\frac{9}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow 16x^2+72x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+72x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (72)^2-4.16.81 & &\\ & = 5184-5184 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-72}{2.16} & & \\ & = -\frac{9}{4} & & \\V &= \Big\{ -\frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(4x^2-(9x-9)=3x(x-5) \\ \Leftrightarrow 4x^2-9x+9=3x^2-15x \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{15}{4}x=-2x^2+\frac{1}{2} \\ \Leftrightarrow 2x^2+\frac{15}{4}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{4.} \left(2x^2+\frac{15}{4}x-\frac{1}{2}\right)=0 \color{red}{.4} \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(2x^2-(9x-16)=x(x-5) \\ \Leftrightarrow 2x^2-9x+16=x^2-5x \\ \Leftrightarrow x^2-4x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.16 & &\\ & = 16-64 & & \\ & = -48 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(x(2x+21)=4(x-2) \\ \Leftrightarrow 2x^2+21x=4x-8 \\ \Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.2.8 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\ & = \frac{-32}{4} & & = \frac{-2}{4} \\ & = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(x(9x+78)=8(x-18) \\ \Leftrightarrow 9x^2+78x=8x-144 \\ \Leftrightarrow 9x^2+70x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+70x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (70)^2-4.9.144 & &\\ & = 4900-5184 & & \\ & = -284 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(5x^2-(18x-1)=x(x-20) \\ \Leftrightarrow 5x^2-18x+1=x^2-20x \\ \Leftrightarrow 4x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.1 & &\\ & = 4-16 & & \\ & = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(8x^2-(9x-6)=7x(x-2) \\ \Leftrightarrow 8x^2-9x+6=7x^2-14x \\ \Leftrightarrow x^2+5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt1}{2.1} & & = \frac{-5+\sqrt1}{2.1} \\ & = \frac{-6}{2} & & = \frac{-4}{2} \\ & = -3 & & = -2 \\ \\ V &= \Big\{ -3 ; -2 \Big\} & &\end{align} \\ -----------------\)