Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(\frac{15}{4}x=-\frac{1}{4}x^2-14\)
- \((-3x-4)(3x-5)-x(-18x+33)=19\)
- \(-(4-38x)=-18x^2-(12-13x)\)
- \(x(x+8)=10(x-10)\)
- \(-(9+5x)=-16x^2-(34-3x)\)
- \(x(x+21)=27(x+1)\)
- \((4x-4)(2x+1)-x(7x+8)=44\)
- \(-\frac{5}{2}x=-\frac{1}{2}x^2-3\)
- \(x(2x+23)=18(x+1)\)
- \(2x^2-(5x+48)=x(x-7)\)
- \(x(x-11)=11(x-11)\)
- \(-\frac{1}{5}x=-\frac{1}{25}x^2+2\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\frac{15}{4}x=-\frac{1}{4}x^2-14 \\
\Leftrightarrow \frac{1}{4}x^2+\frac{15}{4}x+14=0 \\
\Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{15}{4}x+14\right)=0 \color{red}{.4} \\
\Leftrightarrow x^2+15x+56=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+56=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.1.56 & &\\
& = 225-224 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt1}{2.1} & & = \frac{-15+\sqrt1}{2.1} \\
& = \frac{-16}{2} & & = \frac{-14}{2} \\
& = -8 & & = -7 \\ \\ V &= \Big\{ -8 ; -7 \Big\} & &\end{align} \\ -----------------\)
- \((-3x-4)(3x-5)-x(-18x+33)=19\\
\Leftrightarrow -9x^2+15x-12x+20 +18x^2-33x-19=0 \\
\Leftrightarrow 9x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+2x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.9.1 & &\\
& = 4-36 & & \\
& = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(-(4-38x)=-18x^2-(12-13x) \\
\Leftrightarrow -4+38x=-18x^2-12+13x \\
\Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.18.8 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\
& = \frac{-32}{36} & & = \frac{-18}{36} \\
& = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(x(x+8)=10(x-10) \\
\Leftrightarrow x^2+8x=10x-100 \\
\Leftrightarrow x^2-2x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+100=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.1.100 & &\\
& = 4-400 & & \\
& = -396 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(-(9+5x)=-16x^2-(34-3x) \\
\Leftrightarrow -9-5x=-16x^2-34+3x \\
\Leftrightarrow 16x^2-8x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-8x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-8)^2-4.16.25 & &\\
& = 64-1600 & & \\
& = -1536 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x(x+21)=27(x+1) \\
\Leftrightarrow x^2+21x=27x+27 \\
\Leftrightarrow x^2-6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x-27=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-6)^2-4.1.(-27) & &\\
& = 36+108 & & \\
& = 144 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-6)-\sqrt144}{2.1} & & = \frac{-(-6)+\sqrt144}{2.1} \\
& = \frac{-6}{2} & & = \frac{18}{2} \\
& = -3 & & = 9 \\ \\ V &= \Big\{ -3 ; 9 \Big\} & &\end{align} \\ -----------------\)
- \((4x-4)(2x+1)-x(7x+8)=44\\
\Leftrightarrow 8x^2+4x-8x-4 -7x^2-8x-44=0 \\
\Leftrightarrow x^2-8x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x-48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-8)^2-4.1.(-48) & &\\
& = 64+192 & & \\
& = 256 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-8)-\sqrt256}{2.1} & & = \frac{-(-8)+\sqrt256}{2.1} \\
& = \frac{-8}{2} & & = \frac{24}{2} \\
& = -4 & & = 12 \\ \\ V &= \Big\{ -4 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{5}{2}x=-\frac{1}{2}x^2-3 \\
\Leftrightarrow \frac{1}{2}x^2-\frac{5}{2}x+3=0 \\
\Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{5}{2}x+3\right)=0 \color{red}{.2} \\
\Leftrightarrow x^2-5x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x+6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-5)^2-4.1.6 & &\\
& = 25-24 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-5)-\sqrt1}{2.1} & & = \frac{-(-5)+\sqrt1}{2.1} \\
& = \frac{4}{2} & & = \frac{6}{2} \\
& = 2 & & = 3 \\ \\ V &= \Big\{ 2 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(x(2x+23)=18(x+1) \\
\Leftrightarrow 2x^2+23x=18x+18 \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(5x+48)=x(x-7) \\
\Leftrightarrow 2x^2-5x-48=x^2-7x \\
\Leftrightarrow x^2+2x-48=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-48=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.(-48) & &\\
& = 4+192 & & \\
& = 196 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-2-\sqrt196}{2.1} & & = \frac{-2+\sqrt196}{2.1} \\
& = \frac{-16}{2} & & = \frac{12}{2} \\
& = -8 & & = 6 \\ \\ V &= \Big\{ -8 ; 6 \Big\} & &\end{align} \\ -----------------\)
- \(x(x-11)=11(x-11) \\
\Leftrightarrow x^2-11x=11x-121 \\
\Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-22)^2-4.1.121 & &\\
& = 484-484 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-22)}{2.1} & & \\
& = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{1}{5}x=-\frac{1}{25}x^2+2 \\
\Leftrightarrow \frac{1}{25}x^2-\frac{1}{5}x-2=0 \\
\Leftrightarrow \color{red}{25.} \left(\frac{1}{25}x^2-\frac{1}{5}x-2\right)=0 \color{red}{.25} \\
\Leftrightarrow x^2-5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-50=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-5)^2-4.1.(-50) & &\\
& = 25+200 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-5)-\sqrt225}{2.1} & & = \frac{-(-5)+\sqrt225}{2.1} \\
& = \frac{-10}{2} & & = \frac{20}{2} \\
& = -5 & & = 10 \\ \\ V &= \Big\{ -5 ; 10 \Big\} & &\end{align} \\ -----------------\)