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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{7}{10}x=-\frac{1}{5}x^2+\frac{36}{5}\)
2. \(2x^2-(7x-10)=x(x+4)\)
3. \(x(x+36)=32(x+1)\)
4. \(51x^2-(5x+3)=3x(x-4)\)
5. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\)
6. \(x(24x+28)=3(x-2)\)
7. \(x(x-9)=5(x-8)\)
8. \(-(5-8x)=-x^2-(-4-16x)\)
9. \(\frac{1}{3}x^2+\frac{5}{3}x-22=0\)
10. \(5x^2-(10x-144)=x(x+38)\)
11. \(\frac{3}{20}x^2-x+\frac{5}{3}=0\)
12. \(\frac{3}{8}x=-\frac{1}{4}x^2+\frac{1}{4}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{7}{10}x=-\frac{1}{5}x^2+\frac{36}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
2. \(2x^2-(7x-10)=x(x+4) \\ \Leftrightarrow 2x^2-7x+10=x^2+4x \\ \Leftrightarrow x^2-11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.10 & &\\ & = 121-40 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\ & = \frac{2}{2} & & = \frac{20}{2} \\ & = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
3. \(x(x+36)=32(x+1) \\ \Leftrightarrow x^2+36x=32x+32 \\ \Leftrightarrow x^2+4x-32=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-32=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-32) & &\\ & = 16+128 & & \\ & = 144 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt144}{2.1} & & = \frac{-4+\sqrt144}{2.1} \\ & = \frac{-16}{2} & & = \frac{8}{2} \\ & = -8 & & = 4 \\ \\ V &= \Big\{ -8 ; 4 \Big\} & &\end{align} \\ -----------------\)
4. \(51x^2-(5x+3)=3x(x-4) \\ \Leftrightarrow 51x^2-5x-3=3x^2-12x \\ \Leftrightarrow 48x^2+7x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+7x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.48.(-3) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.48} & & = \frac{-7+\sqrt625}{2.48} \\ & = \frac{-32}{96} & & = \frac{18}{96} \\ & = \frac{-1}{3} & & = \frac{3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{3}{16} \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{18}x^2+x+\frac{9}{2}=0\\ \Leftrightarrow \color{red}{18.} \left(\frac{1}{18}x^2+x+\frac{9}{2}\right)=0 \color{red}{.18} \\ \Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-18}{2.1} & & \\ & = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
6. \(x(24x+28)=3(x-2) \\ \Leftrightarrow 24x^2+28x=3x-6 \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(x(x-9)=5(x-8) \\ \Leftrightarrow x^2-9x=5x-40 \\ \Leftrightarrow x^2-14x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+40=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.1.40 & &\\ & = 196-160 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-14)-\sqrt36}{2.1} & & = \frac{-(-14)+\sqrt36}{2.1} \\ & = \frac{8}{2} & & = \frac{20}{2} \\ & = 4 & & = 10 \\ \\ V &= \Big\{ 4 ; 10 \Big\} & &\end{align} \\ -----------------\)
8. \(-(5-8x)=-x^2-(-4-16x) \\ \Leftrightarrow -5+8x=-x^2+4+16x \\ \Leftrightarrow x^2-8x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.(-9) & &\\ & = 64+36 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-8)-\sqrt100}{2.1} & & = \frac{-(-8)+\sqrt100}{2.1} \\ & = \frac{-2}{2} & & = \frac{18}{2} \\ & = -1 & & = 9 \\ \\ V &= \Big\{ -1 ; 9 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{1}{3}x^2+\frac{5}{3}x-22=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{5}{3}x-22\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+5x-66=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+5x-66=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.1.(-66) & &\\ & = 25+264 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt289}{2.1} & & = \frac{-5+\sqrt289}{2.1} \\ & = \frac{-22}{2} & & = \frac{12}{2} \\ & = -11 & & = 6 \\ \\ V &= \Big\{ -11 ; 6 \Big\} & &\end{align} \\ -----------------\)
10. \(5x^2-(10x-144)=x(x+38) \\ \Leftrightarrow 5x^2-10x+144=x^2+38x \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{3}{20}x^2-x+\frac{5}{3}=0\\ \Leftrightarrow \color{red}{60.} \left(\frac{3}{20}x^2-x+\frac{5}{3}\right)=0 \color{red}{.60} \\ \Leftrightarrow 9x^2-60x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-60x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-60)^2-4.9.100 & &\\ & = 3600-3600 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-60)}{2.9} & & \\ & = \frac{10}{3} & & \\V &= \Big\{ \frac{10}{3} \Big\} & &\end{align} \\ -----------------\)
12. \(\frac{3}{8}x=-\frac{1}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{1}{4}x^2+\frac{3}{8}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(\frac{1}{4}x^2+\frac{3}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 2x^2+3x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+3x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.2.(-2) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.2} & & = \frac{-3+\sqrt25}{2.2} \\ & = \frac{-8}{4} & & = \frac{2}{4} \\ & = -2 & & = \frac{1}{2} \\ \\ V &= \Big\{ -2 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)