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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{4}x=-\frac{1}{80}x^2-\frac{5}{4}\)
2. \(8x^2-(9x+4)=4x(x-6)\)
3. \(\frac{7}{24}x=-\frac{1}{2}x^2+\frac{1}{2}\)
4. \(29x^2-(4x-36)=13x(x-4)\)
5. \((5x-4)(4x+5)-x(-4x-20)=-26\)
6. \(\frac{3}{5}x=-\frac{1}{20}x^2-1\)
7. \((-3x+5)(-3x+2)-x(8x+9)=4\)
8. \(\frac{7}{8}x=-9x^2+\frac{1}{4}\)
9. \(-(3-25x)=-6x^2-(9-12x)\)
10. \(18x^2-(20x-33)=17x(x-2)\)
11. \(4x^2-(13x+72)=2x(x-10)\)
12. \(5x^2-(15x-49)=x(x+13)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{4}x=-\frac{1}{80}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{80}x^2+\frac{1}{4}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{80.} \left(\frac{1}{80}x^2+\frac{1}{4}x+\frac{5}{4}\right)=0 \color{red}{.80} \\ \Leftrightarrow x^2+20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-20}{2.1} & & \\ & = -10 & & \\V &= \Big\{ -10 \Big\} & &\end{align} \\ -----------------\)
2. \(8x^2-(9x+4)=4x(x-6) \\ \Leftrightarrow 8x^2-9x-4=4x^2-24x \\ \Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.4.(-4) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\ & = \frac{-32}{8} & & = \frac{2}{8} \\ & = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{7}{24}x=-\frac{1}{2}x^2+\frac{1}{2} \\ \Leftrightarrow \frac{1}{2}x^2+\frac{7}{24}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{24.} \left(\frac{1}{2}x^2+\frac{7}{24}x-\frac{1}{2}\right)=0 \color{red}{.24} \\ \Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.12.(-12) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\ & = \frac{-32}{24} & & = \frac{18}{24} \\ & = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(29x^2-(4x-36)=13x(x-4) \\ \Leftrightarrow 29x^2-4x+36=13x^2-52x \\ \Leftrightarrow 16x^2+48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-48}{2.16} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
5. \((5x-4)(4x+5)-x(-4x-20)=-26\\ \Leftrightarrow 20x^2+25x-16x-20 +4x^2+20x+26=0 \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{3}{5}x=-\frac{1}{20}x^2-1 \\ \Leftrightarrow \frac{1}{20}x^2+\frac{3}{5}x+1=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{20}x^2+\frac{3}{5}x+1\right)=0 \color{red}{.20} \\ \Leftrightarrow x^2+12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.1.20 & &\\ & = 144-80 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-12-\sqrt64}{2.1} & & = \frac{-12+\sqrt64}{2.1} \\ & = \frac{-20}{2} & & = \frac{-4}{2} \\ & = -10 & & = -2 \\ \\ V &= \Big\{ -10 ; -2 \Big\} & &\end{align} \\ -----------------\)
7. \((-3x+5)(-3x+2)-x(8x+9)=4\\ \Leftrightarrow 9x^2-6x-15x+10 -8x^2-9x-4=0 \\ \Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.6 & &\\ & = 25-24 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\ & = \frac{4}{2} & & = \frac{6}{2} \\ & = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{7}{8}x=-9x^2+\frac{1}{4} \\ \Leftrightarrow 9x^2+\frac{7}{8}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(9x^2+\frac{7}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.72.(-2) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\ & = \frac{-32}{144} & & = \frac{18}{144} \\ & = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(-(3-25x)=-6x^2-(9-12x) \\ \Leftrightarrow -3+25x=-6x^2-9+12x \\ \Leftrightarrow 6x^2+13x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+13x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.6.6 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.6} & & = \frac{-13+\sqrt25}{2.6} \\ & = \frac{-18}{12} & & = \frac{-8}{12} \\ & = \frac{-3}{2} & & = \frac{-2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{-2}{3} \Big\} & &\end{align} \\ -----------------\)
10. \(18x^2-(20x-33)=17x(x-2) \\ \Leftrightarrow 18x^2-20x+33=17x^2-34x \\ \Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.1.33 & &\\ & = 196-132 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\ & = \frac{-22}{2} & & = \frac{-6}{2} \\ & = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)
11. \(4x^2-(13x+72)=2x(x-10) \\ \Leftrightarrow 4x^2-13x-72=2x^2-20x \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(5x^2-(15x-49)=x(x+13) \\ \Leftrightarrow 5x^2-15x+49=x^2+13x \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)