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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(59x^2-(19x-3)=11x(x-4)\)
2. \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{9}{2}=0\)
3. \(3x^2-(6x+18)=x(x-11)\)
4. \((-3x-2)(x-3)-x(-7x-29)=-115\)
5. \(10x^2-(19x-2)=8x(x-3)\)
6. \((-x-1)(-x-5)-x(-15x-46)=-44\)
7. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4}\)
8. \(-(4-44x)=-24x^2-(10-19x)\)
9. \(x(x+9)=3(x-3)\)
10. \(\frac{15}{8}x=-x^2+\frac{1}{4}\)
11. \(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}=0\)
12. \((-4x-1)(-2x-4)-x(7x+21)=24\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(59x^2-(19x-3)=11x(x-4) \\ \Leftrightarrow 59x^2-19x+3=11x^2-44x \\ \Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.48.3 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\ & = \frac{-32}{96} & & = \frac{-18}{96} \\ & = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
2. \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{9}{2}=0\\ \Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2+\frac{1}{4}x-\frac{9}{2}\right)=0 \color{red}{.12} \\ \Leftrightarrow x^2+3x-54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-54=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-54) & &\\ & = 9+216 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt225}{2.1} & & = \frac{-3+\sqrt225}{2.1} \\ & = \frac{-18}{2} & & = \frac{12}{2} \\ & = -9 & & = 6 \\ \\ V &= \Big\{ -9 ; 6 \Big\} & &\end{align} \\ -----------------\)
3. \(3x^2-(6x+18)=x(x-11) \\ \Leftrightarrow 3x^2-6x-18=x^2-11x \\ \Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.(-18) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\ & = \frac{-18}{4} & & = \frac{8}{4} \\ & = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
4. \((-3x-2)(x-3)-x(-7x-29)=-115\\ \Leftrightarrow -3x^2+9x-2x+6 +7x^2+29x+115=0 \\ \Leftrightarrow 4x^2+44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-44}{2.4} & & \\ & = -\frac{11}{2} & & \\V &= \Big\{ -\frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
5. \(10x^2-(19x-2)=8x(x-3) \\ \Leftrightarrow 10x^2-19x+2=8x^2-24x \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
6. \((-x-1)(-x-5)-x(-15x-46)=-44\\ \Leftrightarrow x^2+5x+x+5 +15x^2+46x+44=0 \\ \Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (56)^2-4.16.49 & &\\ & = 3136-3136 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-56}{2.16} & & \\ & = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{7}{16}x=-\frac{9}{4}x^2+\frac{1}{4} \\ \Leftrightarrow \frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(-(4-44x)=-24x^2-(10-19x) \\ \Leftrightarrow -4+44x=-24x^2-10+19x \\ \Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.24.6 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\ & = \frac{-32}{48} & & = \frac{-18}{48} \\ & = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
9. \(x(x+9)=3(x-3) \\ \Leftrightarrow x^2+9x=3x-9 \\ \Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.9 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-6}{2.1} & & \\ & = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
10. \(\frac{15}{8}x=-x^2+\frac{1}{4} \\ \Leftrightarrow x^2+\frac{15}{8}x-\frac{1}{4}=0 \\ \Leftrightarrow \color{red}{8.} \left(x^2+\frac{15}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\ \Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.8.(-2) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\ & = \frac{-32}{16} & & = \frac{2}{16} \\ & = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}=0\\ \Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{7}{10}x-\frac{36}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
12. \((-4x-1)(-2x-4)-x(7x+21)=24\\ \Leftrightarrow 8x^2+16x+2x+4 -7x^2-21x-24=0 \\ \Leftrightarrow x^2-x-20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-20) & &\\ & = 1+80 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt81}{2.1} & & = \frac{-(-1)+\sqrt81}{2.1} \\ & = \frac{-8}{2} & & = \frac{10}{2} \\ & = -4 & & = 5 \\ \\ V &= \Big\{ -4 ; 5 \Big\} & &\end{align} \\ -----------------\)