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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(15-11x)=-x^2-(-3-18x)\)
2. \((2x+5)(5x-1)-x(8x-12)=-7\)
3. \(x(x-3)=2(x-2)\)
4. \(-(4-6x)=-x^2-(20-14x)\)
5. \((2x-3)(-x+4)-x(-6x-21)=-16\)
6. \(x=-\frac{8}{5}x^2-\frac{1}{10}\)
7. \(\frac{1}{4}x^2-\frac{13}{18}x+\frac{9}{4}=0\)
8. \(10x^2-(19x-36)=x(x+17)\)
9. \(-\frac{9}{5}x=-\frac{1}{5}x^2-4\)
10. \(-\frac{4}{5}x=-\frac{1}{5}x^2-\frac{3}{5}\)
11. \(x(9x-58)=6(x-24)\)
12. \(5x^2-(8x-16)=x(x+6)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(15-11x)=-x^2-(-3-18x) \\ \Leftrightarrow -15+11x=-x^2+3+18x \\ \Leftrightarrow x^2-7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-7)^2-4.1.(-18) & &\\ & = 49+72 & & \\ & = 121 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-7)-\sqrt121}{2.1} & & = \frac{-(-7)+\sqrt121}{2.1} \\ & = \frac{-4}{2} & & = \frac{18}{2} \\ & = -2 & & = 9 \\ \\ V &= \Big\{ -2 ; 9 \Big\} & &\end{align} \\ -----------------\)
2. \((2x+5)(5x-1)-x(8x-12)=-7\\ \Leftrightarrow 10x^2-2x+25x-5 -8x^2+12x+7=0 \\ \Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.2.2 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\ & = \frac{-8}{4} & & = \frac{-2}{4} \\ & = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
3. \(x(x-3)=2(x-2) \\ \Leftrightarrow x^2-3x=2x-4 \\ \Leftrightarrow x^2-5x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-5)^2-4.1.4 & &\\ & = 25-16 & & \\ & = 9 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-5)-\sqrt9}{2.1} & & = \frac{-(-5)+\sqrt9}{2.1} \\ & = \frac{2}{2} & & = \frac{8}{2} \\ & = 1 & & = 4 \\ \\ V &= \Big\{ 1 ; 4 \Big\} & &\end{align} \\ -----------------\)
4. \(-(4-6x)=-x^2-(20-14x) \\ \Leftrightarrow -4+6x=-x^2-20+14x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
5. \((2x-3)(-x+4)-x(-6x-21)=-16\\ \Leftrightarrow -2x^2+8x+3x-12 +6x^2+21x+16=0 \\ \Leftrightarrow 4x^2+17x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+17x+4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.4.4 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.4} & & = \frac{-17+\sqrt225}{2.4} \\ & = \frac{-32}{8} & & = \frac{-2}{8} \\ & = -4 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -4 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
6. \(x=-\frac{8}{5}x^2-\frac{1}{10} \\ \Leftrightarrow \frac{8}{5}x^2+x+\frac{1}{10}=0 \\ \Leftrightarrow \color{red}{10.} \left(\frac{8}{5}x^2+x+\frac{1}{10}\right)=0 \color{red}{.10} \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{4}x^2-\frac{13}{18}x+\frac{9}{4}=0\\ \Leftrightarrow \color{red}{36.} \left(\frac{1}{4}x^2-\frac{13}{18}x+\frac{9}{4}\right)=0 \color{red}{.36} \\ \Leftrightarrow 9x^2-26x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-26x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-26)^2-4.9.81 & &\\ & = 676-2916 & & \\ & = -2240 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(10x^2-(19x-36)=x(x+17) \\ \Leftrightarrow 10x^2-19x+36=x^2+17x \\ \Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-36)}{2.9} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
9. \(-\frac{9}{5}x=-\frac{1}{5}x^2-4 \\ \Leftrightarrow \frac{1}{5}x^2-\frac{9}{5}x+4=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{9}{5}x+4\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-9x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+20=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-9)^2-4.1.20 & &\\ & = 81-80 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-9)-\sqrt1}{2.1} & & = \frac{-(-9)+\sqrt1}{2.1} \\ & = \frac{8}{2} & & = \frac{10}{2} \\ & = 4 & & = 5 \\ \\ V &= \Big\{ 4 ; 5 \Big\} & &\end{align} \\ -----------------\)
10. \(-\frac{4}{5}x=-\frac{1}{5}x^2-\frac{3}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{4}{5}x+\frac{3}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{4}{5}x+\frac{3}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-4x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-4)^2-4.1.3 & &\\ & = 16-12 & & \\ & = 4 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-4)-\sqrt4}{2.1} & & = \frac{-(-4)+\sqrt4}{2.1} \\ & = \frac{2}{2} & & = \frac{6}{2} \\ & = 1 & & = 3 \\ \\ V &= \Big\{ 1 ; 3 \Big\} & &\end{align} \\ -----------------\)
11. \(x(9x-58)=6(x-24) \\ \Leftrightarrow 9x^2-58x=6x-144 \\ \Leftrightarrow 9x^2-64x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-64x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-64)^2-4.9.144 & &\\ & = 4096-5184 & & \\ & = -1088 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
12. \(5x^2-(8x-16)=x(x+6) \\ \Leftrightarrow 5x^2-8x+16=x^2+6x \\ \Leftrightarrow 4x^2-14x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-14x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-14)^2-4.4.16 & &\\ & = 196-256 & & \\ & = -60 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)