Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(4x=-\frac{8}{11}x^2-\frac{11}{2}\)
- \((3x+2)(4x-4)-x(3x-33)=-12\)
- \(x(2x+23)=18(x+1)\)
- \(2x^2-(15x+96)=x(x-11)\)
- \(-(8+3x)=-x^2-(-36-4x)\)
- \(x(x+14)=3(x-10)\)
- \(17x^2-(9x-9)=x(x+15)\)
- \(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}=0\)
- \(x(9x-10)=2(x-2)\)
- \(x(4x+45)=9(x-9)\)
- \(10x^2-(20x+24)=9x(x-2)\)
- \(-(14-22x)=-x^2-(-85-20x)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(4x=-\frac{8}{11}x^2-\frac{11}{2} \\
\Leftrightarrow \frac{8}{11}x^2+4x+\frac{11}{2}=0 \\
\Leftrightarrow \color{red}{22.} \left(\frac{8}{11}x^2+4x+\frac{11}{2}\right)=0 \color{red}{.22} \\
\Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (88)^2-4.16.121 & &\\
& = 7744-7744 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-88}{2.16} & & \\
& = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
- \((3x+2)(4x-4)-x(3x-33)=-12\\
\Leftrightarrow 12x^2-12x+8x-8 -3x^2+33x+12=0 \\
\Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.9.4 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\
& = \frac{-18}{18} & & = \frac{-8}{18} \\
& = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
- \(x(2x+23)=18(x+1) \\
\Leftrightarrow 2x^2+23x=18x+18 \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(15x+96)=x(x-11) \\
\Leftrightarrow 2x^2-15x-96=x^2-11x \\
\Leftrightarrow x^2-4x-96=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x-96=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.1.(-96) & &\\
& = 16+384 & & \\
& = 400 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-4)-\sqrt400}{2.1} & & = \frac{-(-4)+\sqrt400}{2.1} \\
& = \frac{-16}{2} & & = \frac{24}{2} \\
& = -8 & & = 12 \\ \\ V &= \Big\{ -8 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \(-(8+3x)=-x^2-(-36-4x) \\
\Leftrightarrow -8-3x=-x^2+36+4x \\
\Leftrightarrow x^2-7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-44=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-7)^2-4.1.(-44) & &\\
& = 49+176 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-7)-\sqrt225}{2.1} & & = \frac{-(-7)+\sqrt225}{2.1} \\
& = \frac{-8}{2} & & = \frac{22}{2} \\
& = -4 & & = 11 \\ \\ V &= \Big\{ -4 ; 11 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+14)=3(x-10) \\
\Leftrightarrow x^2+14x=3x-30 \\
\Leftrightarrow x^2+11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+11x+30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (11)^2-4.1.30 & &\\
& = 121-120 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-11-\sqrt1}{2.1} & & = \frac{-11+\sqrt1}{2.1} \\
& = \frac{-12}{2} & & = \frac{-10}{2} \\
& = -6 & & = -5 \\ \\ V &= \Big\{ -6 ; -5 \Big\} & &\end{align} \\ -----------------\)
- \(17x^2-(9x-9)=x(x+15) \\
\Leftrightarrow 17x^2-9x+9=x^2+15x \\
\Leftrightarrow 16x^2-24x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-24x+9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.16.9 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.16} & & \\
& = \frac{3}{4} & & \\V &= \Big\{ \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}=0\\
\Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{19}{5}x+\frac{88}{5}\right)=0 \color{red}{.5} \\
\Leftrightarrow x^2+19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+88=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (19)^2-4.1.88 & &\\
& = 361-352 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-19-\sqrt9}{2.1} & & = \frac{-19+\sqrt9}{2.1} \\
& = \frac{-22}{2} & & = \frac{-16}{2} \\
& = -11 & & = -8 \\ \\ V &= \Big\{ -11 ; -8 \Big\} & &\end{align} \\ -----------------\)
- \(x(9x-10)=2(x-2) \\
\Leftrightarrow 9x^2-10x=2x-4 \\
\Leftrightarrow 9x^2-12x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-12x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-12)^2-4.9.4 & &\\
& = 144-144 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-12)}{2.9} & & \\
& = \frac{2}{3} & & \\V &= \Big\{ \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
- \(x(4x+45)=9(x-9) \\
\Leftrightarrow 4x^2+45x=9x-81 \\
\Leftrightarrow 4x^2+36x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+36x+81=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (36)^2-4.4.81 & &\\
& = 1296-1296 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-36}{2.4} & & \\
& = -\frac{9}{2} & & \\V &= \Big\{ -\frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
- \(10x^2-(20x+24)=9x(x-2) \\
\Leftrightarrow 10x^2-20x-24=9x^2-18x \\
\Leftrightarrow x^2-2x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-24=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.1.(-24) & &\\
& = 4+96 & & \\
& = 100 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-2)-\sqrt100}{2.1} & & = \frac{-(-2)+\sqrt100}{2.1} \\
& = \frac{-8}{2} & & = \frac{12}{2} \\
& = -4 & & = 6 \\ \\ V &= \Big\{ -4 ; 6 \Big\} & &\end{align} \\ -----------------\)
- \(-(14-22x)=-x^2-(-85-20x) \\
\Leftrightarrow -14+22x=-x^2+85+20x \\
\Leftrightarrow x^2+2x-99=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-99=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.(-99) & &\\
& = 4+396 & & \\
& = 400 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-2-\sqrt400}{2.1} & & = \frac{-2+\sqrt400}{2.1} \\
& = \frac{-22}{2} & & = \frac{18}{2} \\
& = -11 & & = 9 \\ \\ V &= \Big\{ -11 ; 9 \Big\} & &\end{align} \\ -----------------\)