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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(2x^2-(18x+18)=x(x-21)\)
2. \(x(x+17)=2(x-25)\)
3. \(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}=0\)
4. \(-(9-31x)=-24x^2-(15-6x)\)
5. \((-5x-1)(-4x-3)-x(19x+14)=-46\)
6. \(\frac{9}{25}x^2-\frac{6}{5}x+1=0\)
7. \((-x-3)(-5x+4)-x(4x-30)=-112\)
8. \(-\frac{1}{3}x=-\frac{1}{12}x^2+8\)
9. \(16x^2-(20x-1)=7x(x-2)\)
10. \(-\frac{7}{4}x=-\frac{1}{12}x^2-9\)
11. \(-2x=-\frac{1}{4}x^2-\frac{25}{4}\)
12. \(\frac{5}{4}x=-\frac{9}{5}x^2-\frac{1}{5}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(2x^2-(18x+18)=x(x-21) \\ \Leftrightarrow 2x^2-18x-18=x^2-21x \\ \Leftrightarrow x^2+3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-18) & &\\ & = 9+72 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt81}{2.1} & & = \frac{-3+\sqrt81}{2.1} \\ & = \frac{-12}{2} & & = \frac{6}{2} \\ & = -6 & & = 3 \\ \\ V &= \Big\{ -6 ; 3 \Big\} & &\end{align} \\ -----------------\)
2. \(x(x+17)=2(x-25) \\ \Leftrightarrow x^2+17x=2x-50 \\ \Leftrightarrow x^2+15x+50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.1.50 & &\\ & = 225-200 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt25}{2.1} & & = \frac{-15+\sqrt25}{2.1} \\ & = \frac{-20}{2} & & = \frac{-10}{2} \\ & = -10 & & = -5 \\ \\ V &= \Big\{ -10 ; -5 \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}=0\\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2+\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow x^2+4x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.4 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.1} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
4. \(-(9-31x)=-24x^2-(15-6x) \\ \Leftrightarrow -9+31x=-24x^2-15+6x \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
5. \((-5x-1)(-4x-3)-x(19x+14)=-46\\ \Leftrightarrow 20x^2+15x+4x+3 -19x^2-14x+46=0 \\ \Leftrightarrow x^2+4x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.49 & &\\ & = 16-196 & & \\ & = -180 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(\frac{9}{25}x^2-\frac{6}{5}x+1=0\\ \Leftrightarrow \color{red}{25.} \left(\frac{9}{25}x^2-\frac{6}{5}x+1\right)=0 \color{red}{.25} \\ \Leftrightarrow 9x^2-30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-30x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-30)^2-4.9.25 & &\\ & = 900-900 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-30)}{2.9} & & \\ & = \frac{5}{3} & & \\V &= \Big\{ \frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
7. \((-x-3)(-5x+4)-x(4x-30)=-112\\ \Leftrightarrow 5x^2-4x+15x-12 -4x^2+30x+112=0 \\ \Leftrightarrow x^2+14x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.100 & &\\ & = 196-400 & & \\ & = -204 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-\frac{1}{3}x=-\frac{1}{12}x^2+8 \\ \Leftrightarrow \frac{1}{12}x^2-\frac{1}{3}x-8=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{1}{3}x-8\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\ & = \frac{-16}{2} & & = \frac{24}{2} \\ & = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
9. \(16x^2-(20x-1)=7x(x-2) \\ \Leftrightarrow 16x^2-20x+1=7x^2-14x \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(-\frac{7}{4}x=-\frac{1}{12}x^2-9 \\ \Leftrightarrow \frac{1}{12}x^2-\frac{7}{4}x+9=0 \\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2-\frac{7}{4}x+9\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2-21x+108=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-21x+108=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-21)^2-4.1.108 & &\\ & = 441-432 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-21)-\sqrt9}{2.1} & & = \frac{-(-21)+\sqrt9}{2.1} \\ & = \frac{18}{2} & & = \frac{24}{2} \\ & = 9 & & = 12 \\ \\ V &= \Big\{ 9 ; 12 \Big\} & &\end{align} \\ -----------------\)
11. \(-2x=-\frac{1}{4}x^2-\frac{25}{4} \\ \Leftrightarrow \frac{1}{4}x^2-2x+\frac{25}{4}=0 \\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-2x+\frac{25}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2-32x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-32x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-32)^2-4.4.100 & &\\ & = 1024-1600 & & \\ & = -576 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(\frac{5}{4}x=-\frac{9}{5}x^2-\frac{1}{5} \\ \Leftrightarrow \frac{9}{5}x^2+\frac{5}{4}x+\frac{1}{5}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{9}{5}x^2+\frac{5}{4}x+\frac{1}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.36.4 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\ & = \frac{-32}{72} & & = \frac{-18}{72} \\ & = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)