Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(65x^2-(9x-3)=17x(x-2)\)
- \(17x^2-(18x-9)=x(x+6)\)
- \(x(3x+29)=4(x-12)\)
- \(\frac{1}{3}x^2-6x+\frac{77}{3}=0\)
- \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}=0\)
- \(x(x+22)=16(x+1)\)
- \(x(16x-77)=11(x-11)\)
- \((-4x-4)(-2x+3)-x(7x-7)=-72\)
- \(9x^2-(9x-18)=x(x-34)\)
- \((x+1)(3x+4)-x(-13x-88)=-140\)
- \((-x-5)(-4x+4)-x(-68x-31)=-18\)
- \(3x^2+\frac{5}{3}x-\frac{4}{3}=0\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(65x^2-(9x-3)=17x(x-2) \\
\Leftrightarrow 65x^2-9x+3=17x^2-34x \\
\Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.48.3 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\
& = \frac{-32}{96} & & = \frac{-18}{96} \\
& = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
- \(17x^2-(18x-9)=x(x+6) \\
\Leftrightarrow 17x^2-18x+9=x^2+6x \\
\Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.16.9 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.16} & & \\
& = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x(3x+29)=4(x-12) \\
\Leftrightarrow 3x^2+29x=4x-48 \\
\Leftrightarrow 3x^2+25x+48=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+25x+48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.3.48 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.3} & & = \frac{-25+\sqrt49}{2.3} \\
& = \frac{-32}{6} & & = \frac{-18}{6} \\
& = \frac{-16}{3} & & = -3 \\ \\ V &= \Big\{ \frac{-16}{3} ; -3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2-6x+\frac{77}{3}=0\\
\Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-6x+\frac{77}{3}\right)=0 \color{red}{.3} \\
\Leftrightarrow x^2-18x+77=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+77=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-18)^2-4.1.77 & &\\
& = 324-308 & & \\
& = 16 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-18)-\sqrt16}{2.1} & & = \frac{-(-18)+\sqrt16}{2.1} \\
& = \frac{14}{2} & & = \frac{22}{2} \\
& = 7 & & = 11 \\ \\ V &= \Big\{ 7 ; 11 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}=0\\
\Leftrightarrow \color{red}{12.} \left(\frac{1}{12}x^2+\frac{1}{4}x-\frac{3}{2}\right)=0 \color{red}{.12} \\
\Leftrightarrow x^2+3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.1.(-18) & &\\
& = 9+72 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt81}{2.1} & & = \frac{-3+\sqrt81}{2.1} \\
& = \frac{-12}{2} & & = \frac{6}{2} \\
& = -6 & & = 3 \\ \\ V &= \Big\{ -6 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+22)=16(x+1) \\
\Leftrightarrow x^2+22x=16x+16 \\
\Leftrightarrow x^2+6x-16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.1.(-16) & &\\
& = 36+64 & & \\
& = 100 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-6-\sqrt100}{2.1} & & = \frac{-6+\sqrt100}{2.1} \\
& = \frac{-16}{2} & & = \frac{4}{2} \\
& = -8 & & = 2 \\ \\ V &= \Big\{ -8 ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(x(16x-77)=11(x-11) \\
\Leftrightarrow 16x^2-77x=11x-121 \\
\Leftrightarrow 16x^2-88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-88x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-88)^2-4.16.121 & &\\
& = 7744-7744 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-88)}{2.16} & & \\
& = \frac{11}{4} & & \\V &= \Big\{ \frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
- \((-4x-4)(-2x+3)-x(7x-7)=-72\\
\Leftrightarrow 8x^2-12x+8x-12 -7x^2+7x+72=0 \\
\Leftrightarrow x^2-17x+60=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+60=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-17)^2-4.1.60 & &\\
& = 289-240 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-17)-\sqrt49}{2.1} & & = \frac{-(-17)+\sqrt49}{2.1} \\
& = \frac{10}{2} & & = \frac{24}{2} \\
& = 5 & & = 12 \\ \\ V &= \Big\{ 5 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \(9x^2-(9x-18)=x(x-34) \\
\Leftrightarrow 9x^2-9x+18=x^2-34x \\
\Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.8.18 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\
& = \frac{-32}{16} & & = \frac{-18}{16} \\
& = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
- \((x+1)(3x+4)-x(-13x-88)=-140\\
\Leftrightarrow 3x^2+4x+3x+4 +13x^2+88x+140=0 \\
\Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (96)^2-4.16.144 & &\\
& = 9216-9216 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-96}{2.16} & & \\
& = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
- \((-x-5)(-4x+4)-x(-68x-31)=-18\\
\Leftrightarrow 4x^2-4x+20x-20 +68x^2+31x+18=0 \\
\Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.72.(-2) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\
& = \frac{-32}{144} & & = \frac{18}{144} \\
& = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(3x^2+\frac{5}{3}x-\frac{4}{3}=0\\
\Leftrightarrow \color{red}{3.} \left(3x^2+\frac{5}{3}x-\frac{4}{3}\right)=0 \color{red}{.3} \\
\Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.9.(-4) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\
& = \frac{-18}{18} & & = \frac{8}{18} \\
& = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)