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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{2}x^2+\frac{15}{4}x-2=0\)
2. \((-2x+3)(-4x+3)-x(-8x-37)=-16\)
3. \((-3x+4)(5x-2)-x(-19x+26)=-57\)
4. \(\frac{4}{25}x^2+\frac{4}{5}x+1=0\)
5. \(10x^2-(7x-100)=x(x-31)\)
6. \(2x^2-(10x-20)=x(x+2)\)
7. \(\frac{9}{5}x^2+\frac{7}{10}x-\frac{4}{5}=0\)
8. \(x(x+1)=-4(x+1)\)
9. \(\frac{7}{8}x=-\frac{9}{2}x^2+\frac{1}{2}\)
10. \(-(2-20x)=-3x^2-(-10-15x)\)
11. \(-\frac{1}{4}x=-\frac{1}{80}x^2-\frac{5}{4}\)
12. \((5x-5)(4x+2)-x(16x+4)=-11\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{2}x^2+\frac{15}{4}x-2=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{15}{4}x-2\right)=0 \color{red}{.4} \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \((-2x+3)(-4x+3)-x(-8x-37)=-16\\ \Leftrightarrow 8x^2-6x-12x+9 +8x^2+37x+16=0 \\ \Leftrightarrow 16x^2+40x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+40x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (40)^2-4.16.25 & &\\ & = 1600-1600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-40}{2.16} & & \\ & = -\frac{5}{4} & & \\V &= \Big\{ -\frac{5}{4} \Big\} & &\end{align} \\ -----------------\)
3. \((-3x+4)(5x-2)-x(-19x+26)=-57\\ \Leftrightarrow -15x^2+6x+20x-8 +19x^2-26x+57=0 \\ \Leftrightarrow 4x^2-28x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-28)^2-4.4.49 & &\\ & = 784-784 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-28)}{2.4} & & \\ & = \frac{7}{2} & & \\V &= \Big\{ \frac{7}{2} \Big\} & &\end{align} \\ -----------------\)
4. \(\frac{4}{25}x^2+\frac{4}{5}x+1=0\\ \Leftrightarrow \color{red}{25.} \left(\frac{4}{25}x^2+\frac{4}{5}x+1\right)=0 \color{red}{.25} \\ \Leftrightarrow 16x^2+80x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+80x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (80)^2-4.16.100 & &\\ & = 6400-6400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-80}{2.16} & & \\ & = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(10x^2-(7x-100)=x(x-31) \\ \Leftrightarrow 10x^2-7x+100=x^2-31x \\ \Leftrightarrow 9x^2+24x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+24x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (24)^2-4.9.100 & &\\ & = 576-3600 & & \\ & = -3024 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(2x^2-(10x-20)=x(x+2) \\ \Leftrightarrow 2x^2-10x+20=x^2+2x \\ \Leftrightarrow x^2-12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.20 & &\\ & = 144-80 & & \\ & = 64 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-12)-\sqrt64}{2.1} & & = \frac{-(-12)+\sqrt64}{2.1} \\ & = \frac{4}{2} & & = \frac{20}{2} \\ & = 2 & & = 10 \\ \\ V &= \Big\{ 2 ; 10 \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{9}{5}x^2+\frac{7}{10}x-\frac{4}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{9}{5}x^2+\frac{7}{10}x-\frac{4}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(x(x+1)=-4(x+1) \\ \Leftrightarrow x^2+x=-4x-4 \\ \Leftrightarrow x^2+5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.1} & & = \frac{-5+\sqrt9}{2.1} \\ & = \frac{-8}{2} & & = \frac{-2}{2} \\ & = -4 & & = -1 \\ \\ V &= \Big\{ -4 ; -1 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{7}{8}x=-\frac{9}{2}x^2+\frac{1}{2} \\ \Leftrightarrow \frac{9}{2}x^2+\frac{7}{8}x-\frac{1}{2}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{9}{2}x^2+\frac{7}{8}x-\frac{1}{2}\right)=0 \color{red}{.8} \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(-(2-20x)=-3x^2-(-10-15x) \\ \Leftrightarrow -2+20x=-3x^2+10+15x \\ \Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.3.(-12) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\ & = \frac{-18}{6} & & = \frac{8}{6} \\ & = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(-\frac{1}{4}x=-\frac{1}{80}x^2-\frac{5}{4} \\ \Leftrightarrow \frac{1}{80}x^2-\frac{1}{4}x+\frac{5}{4}=0 \\ \Leftrightarrow \color{red}{80.} \left(\frac{1}{80}x^2-\frac{1}{4}x+\frac{5}{4}\right)=0 \color{red}{.80} \\ \Leftrightarrow x^2-20x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-20x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.1.100 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.1} & & \\ & = 10 & & \\V &= \Big\{ 10 \Big\} & &\end{align} \\ -----------------\)
12. \((5x-5)(4x+2)-x(16x+4)=-11\\ \Leftrightarrow 20x^2+10x-20x-10 -16x^2-4x+11=0 \\ \Leftrightarrow 4x^2-4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-4)}{2.4} & & \\ & = \frac{1}{2} & & \\V &= \Big\{ \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)