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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-\frac{1}{4}x=-\frac{1}{48}x^2-\frac{3}{4}\)
2. \(-(5-10x)=-4x^2-(41-8x)\)
3. \(11x^2-(16x-144)=2x(x-35)\)
4. \(17x^2-(14x-4)=x(x-26)\)
5. \(x(x+11)=5(x-5)\)
6. \(-(4-30x)=-4x^2-(13-17x)\)
7. \(5x^2-(11x-121)=x(x+33)\)
8. \(4x^2-(2x-121)=3x(x-6)\)
9. \(\frac{5}{4}x=-\frac{1}{5}x^2-\frac{9}{5}\)
10. \((2x-2)(x-5)-x(-2x-3)=11\)
11. \(\frac{9}{4}x^2+\frac{5}{2}x+1=0\)
12. \(-\frac{23}{5}x=-\frac{9}{10}x^2-\frac{32}{5}\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-\frac{1}{4}x=-\frac{1}{48}x^2-\frac{3}{4} \\ \Leftrightarrow \frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}=0 \\ \Leftrightarrow \color{red}{48.} \left(\frac{1}{48}x^2-\frac{1}{4}x+\frac{3}{4}\right)=0 \color{red}{.48} \\ \Leftrightarrow 4x^2-48x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-48x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.4.144 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.4} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
2. \(-(5-10x)=-4x^2-(41-8x) \\ \Leftrightarrow -5+10x=-4x^2-41+8x \\ \Leftrightarrow 4x^2+2x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.36 & &\\ & = 4-576 & & \\ & = -572 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(11x^2-(16x-144)=2x(x-35) \\ \Leftrightarrow 11x^2-16x+144=2x^2-70x \\ \Leftrightarrow 9x^2+54x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+54x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (54)^2-4.9.144 & &\\ & = 2916-5184 & & \\ & = -2268 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(17x^2-(14x-4)=x(x-26) \\ \Leftrightarrow 17x^2-14x+4=x^2-26x \\ \Leftrightarrow 16x^2+12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+12x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.16.4 & &\\ & = 144-256 & & \\ & = -112 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x(x+11)=5(x-5) \\ \Leftrightarrow x^2+11x=5x-25 \\ \Leftrightarrow x^2+6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.1.25 & &\\ & = 36-100 & & \\ & = -64 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(-(4-30x)=-4x^2-(13-17x) \\ \Leftrightarrow -4+30x=-4x^2-13+17x \\ \Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.4.9 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\ & = \frac{-18}{8} & & = \frac{-8}{8} \\ & = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
7. \(5x^2-(11x-121)=x(x+33) \\ \Leftrightarrow 5x^2-11x+121=x^2+33x \\ \Leftrightarrow 4x^2-44x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-44x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-44)^2-4.4.121 & &\\ & = 1936-1936 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-44)}{2.4} & & \\ & = \frac{11}{2} & & \\V &= \Big\{ \frac{11}{2} \Big\} & &\end{align} \\ -----------------\)
8. \(4x^2-(2x-121)=3x(x-6) \\ \Leftrightarrow 4x^2-2x+121=3x^2-18x \\ \Leftrightarrow x^2+16x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (16)^2-4.1.121 & &\\ & = 256-484 & & \\ & = -228 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \(\frac{5}{4}x=-\frac{1}{5}x^2-\frac{9}{5} \\ \Leftrightarrow \frac{1}{5}x^2+\frac{5}{4}x+\frac{9}{5}=0 \\ \Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+\frac{5}{4}x+\frac{9}{5}\right)=0 \color{red}{.20} \\ \Leftrightarrow 4x^2+25x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+25x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.4.36 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.4} & & = \frac{-25+\sqrt49}{2.4} \\ & = \frac{-32}{8} & & = \frac{-18}{8} \\ & = -4 & & = \frac{-9}{4} \\ \\ V &= \Big\{ -4 ; \frac{-9}{4} \Big\} & &\end{align} \\ -----------------\)
10. \((2x-2)(x-5)-x(-2x-3)=11\\ \Leftrightarrow 2x^2-10x-2x+10 +2x^2+3x-11=0 \\ \Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.4.(-1) & &\\ & = 9+16 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\ & = \frac{-8}{8} & & = \frac{2}{8} \\ & = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
11. \(\frac{9}{4}x^2+\frac{5}{2}x+1=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{9}{4}x^2+\frac{5}{2}x+1\right)=0 \color{red}{.4} \\ \Leftrightarrow 9x^2+10x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+10x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.9.4 & &\\ & = 100-144 & & \\ & = -44 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(-\frac{23}{5}x=-\frac{9}{10}x^2-\frac{32}{5} \\ \Leftrightarrow \frac{9}{10}x^2-\frac{23}{5}x+\frac{32}{5}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{9}{10}x^2-\frac{23}{5}x+\frac{32}{5}\right)=0 \color{red}{.10} \\ \Leftrightarrow 9x^2-46x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-46x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-46)^2-4.9.64 & &\\ & = 2116-2304 & & \\ & = -188 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)