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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(12+3x)=-16x^2-(37-17x)\)
2. \(-(13-17x)=-4x^2-(22-4x)\)
3. \(\frac{2}{3}x^2+\frac{7}{3}x+\frac{49}{24}=0\)
4. \(-(10-24x)=-x^2-(59-10x)\)
5. \(\frac{9}{8}x^2-\frac{3}{2}x+\frac{1}{2}=0\)
6. \((-x+3)(5x-3)-x(-6x+10)=-69\)
7. \(x(16x-31)=-9(x+1)\)
8. \(\frac{1}{5}x^2+\frac{28}{45}x+\frac{4}{5}=0\)
9. \((-5x-5)(3x-2)-x(-19x-8)=-39\)
10. \(\frac{1}{5}x^2-\frac{4}{5}x+\frac{81}{5}=0\)
11. \(2x^2-(2x-16)=x(x+6)\)
12. \(\frac{1}{8}x^2-\frac{3}{4}x+1=0\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(12+3x)=-16x^2-(37-17x) \\ \Leftrightarrow -12-3x=-16x^2-37+17x \\ \Leftrightarrow 16x^2-20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.16.25 & &\\ & = 400-1600 & & \\ & = -1200 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(-(13-17x)=-4x^2-(22-4x) \\ \Leftrightarrow -13+17x=-4x^2-22+4x \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{2}{3}x^2+\frac{7}{3}x+\frac{49}{24}=0\\ \Leftrightarrow \color{red}{24.} \left(\frac{2}{3}x^2+\frac{7}{3}x+\frac{49}{24}\right)=0 \color{red}{.24} \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(-(10-24x)=-x^2-(59-10x) \\ \Leftrightarrow -10+24x=-x^2-59+10x \\ \Leftrightarrow x^2+14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.49 & &\\ & = 196-196 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-14}{2.1} & & \\ & = -7 & & \\V &= \Big\{ -7 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{9}{8}x^2-\frac{3}{2}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{8}x^2-\frac{3}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.4 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.9} & & \\ & = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
6. \((-x+3)(5x-3)-x(-6x+10)=-69\\ \Leftrightarrow -5x^2+3x+15x-9 +6x^2-10x+69=0 \\ \Leftrightarrow x^2-16x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.1.60 & &\\ & = 256-240 & & \\ & = 16 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-16)-\sqrt16}{2.1} & & = \frac{-(-16)+\sqrt16}{2.1} \\ & = \frac{12}{2} & & = \frac{20}{2} \\ & = 6 & & = 10 \\ \\ V &= \Big\{ 6 ; 10 \Big\} & &\end{align} \\ -----------------\)
7. \(x(16x-31)=-9(x+1) \\ \Leftrightarrow 16x^2-31x=-9x-9 \\ \Leftrightarrow 16x^2-22x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-22x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-22)^2-4.16.9 & &\\ & = 484-576 & & \\ & = -92 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+\frac{28}{45}x+\frac{4}{5}=0\\ \Leftrightarrow \color{red}{45.} \left(\frac{1}{5}x^2+\frac{28}{45}x+\frac{4}{5}\right)=0 \color{red}{.45} \\ \Leftrightarrow 9x^2+28x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+28x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.9.36 & &\\ & = 784-1296 & & \\ & = -512 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \((-5x-5)(3x-2)-x(-19x-8)=-39\\ \Leftrightarrow -15x^2+10x-15x+10 +19x^2+8x+39=0 \\ \Leftrightarrow 4x^2+28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-28}{2.4} & & \\ & = -\frac{7}{2} & & \\V &= \Big\{ -\frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{1}{5}x^2-\frac{4}{5}x+\frac{81}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{4}{5}x+\frac{81}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-4x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.81 & &\\ & = 16-324 & & \\ & = -308 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(2x^2-(2x-16)=x(x+6) \\ \Leftrightarrow 2x^2-2x+16=x^2+6x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{1}{8}x^2-\frac{3}{4}x+1=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{3}{4}x+1\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2-6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-6)-\sqrt4}{2.1} & & = \frac{-(-6)+\sqrt4}{2.1} \\ & = \frac{4}{2} & & = \frac{8}{2} \\ & = 2 & & = 4 \\ \\ V &= \Big\{ 2 ; 4 \Big\} & &\end{align} \\ -----------------\)