Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(-(4-29x)=-x^2-(31-17x)\)
- \(\frac{3}{4}x=-\frac{1}{16}x^2-\frac{5}{4}\)
- \(x(16x-33)=-121(x+1)\)
- \(x(9x+15)=2(x-2)\)
- \((-3x-2)(-5x-2)-x(-3x+3)=12\)
- \(3x^2-(6x+18)=x(x-11)\)
- \(9x^2-(14x-2)=x(x-31)\)
- \(-(13-32x)=-2x^2-(5-17x)\)
- \((-5x+5)(4x+4)-x(-36x-10)=19\)
- \(x(4x+25)=5(x-5)\)
- \(2x^2-(20x-81)=x(x-2)\)
- \(-\frac{1}{10}x=-\frac{1}{5}x^2-\frac{4}{5}\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(-(4-29x)=-x^2-(31-17x) \\
\Leftrightarrow -4+29x=-x^2-31+17x \\
\Leftrightarrow x^2+12x+27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+27=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (12)^2-4.1.27 & &\\
& = 144-108 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-12-\sqrt36}{2.1} & & = \frac{-12+\sqrt36}{2.1} \\
& = \frac{-18}{2} & & = \frac{-6}{2} \\
& = -9 & & = -3 \\ \\ V &= \Big\{ -9 ; -3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{3}{4}x=-\frac{1}{16}x^2-\frac{5}{4} \\
\Leftrightarrow \frac{1}{16}x^2+\frac{3}{4}x+\frac{5}{4}=0 \\
\Leftrightarrow \color{red}{16.} \left(\frac{1}{16}x^2+\frac{3}{4}x+\frac{5}{4}\right)=0 \color{red}{.16} \\
\Leftrightarrow x^2+12x+20=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+20=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (12)^2-4.1.20 & &\\
& = 144-80 & & \\
& = 64 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-12-\sqrt64}{2.1} & & = \frac{-12+\sqrt64}{2.1} \\
& = \frac{-20}{2} & & = \frac{-4}{2} \\
& = -10 & & = -2 \\ \\ V &= \Big\{ -10 ; -2 \Big\} & &\end{align} \\ -----------------\)
- \(x(16x-33)=-121(x+1) \\
\Leftrightarrow 16x^2-33x=-121x-121 \\
\Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (88)^2-4.16.121 & &\\
& = 7744-7744 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-88}{2.16} & & \\
& = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x(9x+15)=2(x-2) \\
\Leftrightarrow 9x^2+15x=2x-4 \\
\Leftrightarrow 9x^2+13x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+13x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.9.4 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.9} & & = \frac{-13+\sqrt25}{2.9} \\
& = \frac{-18}{18} & & = \frac{-8}{18} \\
& = -1 & & = \frac{-4}{9} \\ \\ V &= \Big\{ -1 ; \frac{-4}{9} \Big\} & &\end{align} \\ -----------------\)
- \((-3x-2)(-5x-2)-x(-3x+3)=12\\
\Leftrightarrow 15x^2+6x+10x+4 +3x^2-3x-12=0 \\
\Leftrightarrow 18x^2+7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+7x-8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.18.(-8) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.18} & & = \frac{-7+\sqrt625}{2.18} \\
& = \frac{-32}{36} & & = \frac{18}{36} \\
& = \frac{-8}{9} & & = \frac{1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(6x+18)=x(x-11) \\
\Leftrightarrow 3x^2-6x-18=x^2-11x \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(9x^2-(14x-2)=x(x-31) \\
\Leftrightarrow 9x^2-14x+2=x^2-31x \\
\Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.8.2 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\
& = \frac{-32}{16} & & = \frac{-2}{16} \\
& = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(-(13-32x)=-2x^2-(5-17x) \\
\Leftrightarrow -13+32x=-2x^2-5+17x \\
\Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.2.(-8) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\
& = \frac{-32}{4} & & = \frac{2}{4} \\
& = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
- \((-5x+5)(4x+4)-x(-36x-10)=19\\
\Leftrightarrow -20x^2-20x+20x+20 +36x^2+10x-19=0 \\
\Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (10)^2-4.16.1 & &\\
& = 100-64 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\
& = \frac{-16}{32} & & = \frac{-4}{32} \\
& = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(x(4x+25)=5(x-5) \\
\Leftrightarrow 4x^2+25x=5x-25 \\
\Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (20)^2-4.4.25 & &\\
& = 400-400 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-20}{2.4} & & \\
& = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(20x-81)=x(x-2) \\
\Leftrightarrow 2x^2-20x+81=x^2-2x \\
\Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-18)^2-4.1.81 & &\\
& = 324-324 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-18)}{2.1} & & \\
& = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{1}{10}x=-\frac{1}{5}x^2-\frac{4}{5} \\
\Leftrightarrow \frac{1}{5}x^2-\frac{1}{10}x+\frac{4}{5}=0 \\
\Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2-\frac{1}{10}x+\frac{4}{5}\right)=0 \color{red}{.10} \\
\Leftrightarrow 4x^2-2x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.4.16 & &\\
& = 4-256 & & \\
& = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)