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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(x-8)=4(x-9)\)
2. \((2x-2)(-2x-2)-x(-8x-14)=-117\)
3. \(10x^2-(6x-36)=x(x-42)\)
4. \(-(13-42x)=-x^2-(145-19x)\)
5. \(\frac{1}{9}x^2+\frac{1}{3}x-2=0\)
6. \(x(18x+15)=8(x+1)\)
7. \(\frac{1}{3}x^2+\frac{17}{3}x+20=0\)
8. \(\frac{1}{5}x^2+\frac{4}{5}x-1=0\)
9. \(\frac{16}{3}x^2+5x-\frac{1}{3}=0\)
10. \(x(4x+15)=3(x-3)\)
11. \(-\frac{4}{3}x=-\frac{4}{15}x^2-\frac{5}{3}\)
12. \((4x+2)(2x-3)-x(-x-6)=-31\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(x-8)=4(x-9) \\ \Leftrightarrow x^2-8x=4x-36 \\ \Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.1.36 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-12)}{2.1} & & \\ & = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
2. \((2x-2)(-2x-2)-x(-8x-14)=-117\\ \Leftrightarrow -4x^2-4x+4x+4 +8x^2+14x+117=0 \\ \Leftrightarrow 4x^2+14x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+14x+121=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (14)^2-4.4.121 & &\\ & = 196-1936 & & \\ & = -1740 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(10x^2-(6x-36)=x(x-42) \\ \Leftrightarrow 10x^2-6x+36=x^2-42x \\ \Leftrightarrow 9x^2+36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+36x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (36)^2-4.9.36 & &\\ & = 1296-1296 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-36}{2.9} & & \\ & = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
4. \(-(13-42x)=-x^2-(145-19x) \\ \Leftrightarrow -13+42x=-x^2-145+19x \\ \Leftrightarrow x^2+23x+132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+23x+132=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (23)^2-4.1.132 & &\\ & = 529-528 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-23-\sqrt1}{2.1} & & = \frac{-23+\sqrt1}{2.1} \\ & = \frac{-24}{2} & & = \frac{-22}{2} \\ & = -12 & & = -11 \\ \\ V &= \Big\{ -12 ; -11 \Big\} & &\end{align} \\ -----------------\)
5. \(\frac{1}{9}x^2+\frac{1}{3}x-2=0\\ \Leftrightarrow \color{red}{9.} \left(\frac{1}{9}x^2+\frac{1}{3}x-2\right)=0 \color{red}{.9} \\ \Leftrightarrow x^2+3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-18=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (3)^2-4.1.(-18) & &\\ & = 9+72 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-3-\sqrt81}{2.1} & & = \frac{-3+\sqrt81}{2.1} \\ & = \frac{-12}{2} & & = \frac{6}{2} \\ & = -6 & & = 3 \\ \\ V &= \Big\{ -6 ; 3 \Big\} & &\end{align} \\ -----------------\)
6. \(x(18x+15)=8(x+1) \\ \Leftrightarrow 18x^2+15x=8x+8 \\ \Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.18.(-8) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\ & = \frac{-32}{36} & & = \frac{18}{36} \\ & = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{3}x^2+\frac{17}{3}x+20=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2+\frac{17}{3}x+20\right)=0 \color{red}{.3} \\ \Leftrightarrow x^2+17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+60=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.1.60 & &\\ & = 289-240 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt49}{2.1} & & = \frac{-17+\sqrt49}{2.1} \\ & = \frac{-24}{2} & & = \frac{-10}{2} \\ & = -12 & & = -5 \\ \\ V &= \Big\{ -12 ; -5 \Big\} & &\end{align} \\ -----------------\)
8. \(\frac{1}{5}x^2+\frac{4}{5}x-1=0\\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{4}{5}x-1\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2+4x-5=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x-5=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.1.(-5) & &\\ & = 16+20 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-4-\sqrt36}{2.1} & & = \frac{-4+\sqrt36}{2.1} \\ & = \frac{-10}{2} & & = \frac{2}{2} \\ & = -5 & & = 1 \\ \\ V &= \Big\{ -5 ; 1 \Big\} & &\end{align} \\ -----------------\)
9. \(\frac{16}{3}x^2+5x-\frac{1}{3}=0\\ \Leftrightarrow \color{red}{3.} \left(\frac{16}{3}x^2+5x-\frac{1}{3}\right)=0 \color{red}{.3} \\ \Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.16.(-1) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\ & = \frac{-32}{32} & & = \frac{2}{32} \\ & = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
10. \(x(4x+15)=3(x-3) \\ \Leftrightarrow 4x^2+15x=3x-9 \\ \Leftrightarrow 4x^2+12x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+12x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (12)^2-4.4.9 & &\\ & = 144-144 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-12}{2.4} & & \\ & = -\frac{3}{2} & & \\V &= \Big\{ -\frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
11. \(-\frac{4}{3}x=-\frac{4}{15}x^2-\frac{5}{3} \\ \Leftrightarrow \frac{4}{15}x^2-\frac{4}{3}x+\frac{5}{3}=0 \\ \Leftrightarrow \color{red}{15.} \left(\frac{4}{15}x^2-\frac{4}{3}x+\frac{5}{3}\right)=0 \color{red}{.15} \\ \Leftrightarrow 4x^2-20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-20x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-20)^2-4.4.25 & &\\ & = 400-400 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-20)}{2.4} & & \\ & = \frac{5}{2} & & \\V &= \Big\{ \frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
12. \((4x+2)(2x-3)-x(-x-6)=-31\\ \Leftrightarrow 8x^2-12x+4x-6 +x^2+6x+31=0 \\ \Leftrightarrow 9x^2-12x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.9.25 & &\\ & = 144-900 & & \\ & = -756 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)