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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(15-22x)=-x^2-(-81-18x)\)
2. \(\frac{9}{22}x^2+3x+\frac{11}{2}=0\)
3. \(x^2+\frac{5}{4}x+\frac{1}{4}=0\)
4. \(x=-\frac{1}{6}x^2-\frac{4}{3}\)
5. \(x(18x+7)=2(x+1)\)
6. \(\frac{16}{5}x^2+3x-\frac{1}{5}=0\)
7. \(-(7-21x)=-24x^2-(1-14x)\)
8. \(11x^2-(10x-36)=7x(x-4)\)
9. \((x-4)(-4x-1)-x(-8x-4)=40\)
10. \(-(13-58x)=-4x^2-(134-14x)\)
11. \(2x^2-(19x+21)=x(x-23)\)
12. \(-(6-10x)=-x^2-(9-14x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(15-22x)=-x^2-(-81-18x) \\ \Leftrightarrow -15+22x=-x^2+81+18x \\ \Leftrightarrow x^2+4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-96=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-96) & &\\ & = 16+384 & & \\ & = 400 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt400}{2.1} & & = \frac{-4+\sqrt400}{2.1} \\ & = \frac{-24}{2} & & = \frac{16}{2} \\ & = -12 & & = 8 \\ \\ V &= \Big\{ -12 ; 8 \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{9}{22}x^2+3x+\frac{11}{2}=0\\ \Leftrightarrow \color{red}{22.} \left(\frac{9}{22}x^2+3x+\frac{11}{2}\right)=0 \color{red}{.22} \\ \Leftrightarrow 9x^2+66x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+66x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (66)^2-4.9.121 & &\\ & = 4356-4356 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-66}{2.9} & & \\ & = -\frac{11}{3} & & \\V &= \Big\{ -\frac{11}{3} \Big\} & &\end{align} \\ -----------------\)
3. \(x^2+\frac{5}{4}x+\frac{1}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(x^2+\frac{5}{4}x+\frac{1}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.1 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\ & = \frac{-8}{8} & & = \frac{-2}{8} \\ & = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
4. \(x=-\frac{1}{6}x^2-\frac{4}{3} \\ \Leftrightarrow \frac{1}{6}x^2+x+\frac{4}{3}=0 \\ \Leftrightarrow \color{red}{6.} \left(\frac{1}{6}x^2+x+\frac{4}{3}\right)=0 \color{red}{.6} \\ \Leftrightarrow x^2+6x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.8 & &\\ & = 36-32 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt4}{2.1} & & = \frac{-6+\sqrt4}{2.1} \\ & = \frac{-8}{2} & & = \frac{-4}{2} \\ & = -4 & & = -2 \\ \\ V &= \Big\{ -4 ; -2 \Big\} & &\end{align} \\ -----------------\)
5. \(x(18x+7)=2(x+1) \\ \Leftrightarrow 18x^2+7x=2x+2 \\ \Leftrightarrow 18x^2+5x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+5x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.18.(-2) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.18} & & = \frac{-5+\sqrt169}{2.18} \\ & = \frac{-18}{36} & & = \frac{8}{36} \\ & = \frac{-1}{2} & & = \frac{2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{2}{9} \Big\} & &\end{align} \\ -----------------\)
6. \(\frac{16}{5}x^2+3x-\frac{1}{5}=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{16}{5}x^2+3x-\frac{1}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
7. \(-(7-21x)=-24x^2-(1-14x) \\ \Leftrightarrow -7+21x=-24x^2-1+14x \\ \Leftrightarrow 24x^2+7x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+7x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.24.(-6) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.24} & & = \frac{-7+\sqrt625}{2.24} \\ & = \frac{-32}{48} & & = \frac{18}{48} \\ & = \frac{-2}{3} & & = \frac{3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{3}{8} \Big\} & &\end{align} \\ -----------------\)
8. \(11x^2-(10x-36)=7x(x-4) \\ \Leftrightarrow 11x^2-10x+36=7x^2-28x \\ \Leftrightarrow 4x^2+18x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+18x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (18)^2-4.4.36 & &\\ & = 324-576 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \((x-4)(-4x-1)-x(-8x-4)=40\\ \Leftrightarrow -4x^2-x+16x+4 +8x^2+4x-40=0 \\ \Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.4.(-36) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\ & = \frac{-32}{8} & & = \frac{18}{8} \\ & = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
10. \(-(13-58x)=-4x^2-(134-14x) \\ \Leftrightarrow -13+58x=-4x^2-134+14x \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(2x^2-(19x+21)=x(x-23) \\ \Leftrightarrow 2x^2-19x-21=x^2-23x \\ \Leftrightarrow x^2+4x-21=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-21=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-21) & &\\ & = 16+84 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt100}{2.1} & & = \frac{-4+\sqrt100}{2.1} \\ & = \frac{-14}{2} & & = \frac{6}{2} \\ & = -7 & & = 3 \\ \\ V &= \Big\{ -7 ; 3 \Big\} & &\end{align} \\ -----------------\)
12. \(-(6-10x)=-x^2-(9-14x) \\ \Leftrightarrow -6+10x=-x^2-9+14x \\ \Leftrightarrow x^2-4x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt4}{2.1} & & = \frac{-(-4)+\sqrt4}{2.1} \\ & = \frac{2}{2} & & = \frac{6}{2} \\ & = 1 & & = 3 \\ \\ V &= \Big\{ 1 ; 3 \Big\} & &\end{align} \\ -----------------\)