Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(-(6-15x)=-x^2-(46-2x)\)
- \(\frac{1}{4}x^2-\frac{1}{2}x+\frac{16}{9}=0\)
- \(-(14-10x)=-x^2-(4-7x)\)
- \((4x+4)(-5x-4)-x(-92x-57)=-18\)
- \((x-5)(5x-2)-x(4x+24)=-54\)
- \(x(4x+1)=-(x+1)\)
- \(\frac{5}{3}x=-3x^2+\frac{4}{3}\)
- \(5x^2-(6x+36)=x(x-13)\)
- \(-(14-45x)=-9x^2-(39-15x)\)
- \(\frac{1}{2}x^2+\frac{19}{2}x+45=0\)
- \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2}\)
- \(x(16x+9)=-(x+1)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(-(6-15x)=-x^2-(46-2x) \\
\Leftrightarrow -6+15x=-x^2-46+2x \\
\Leftrightarrow x^2+13x+40=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+40=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.1.40 & &\\
& = 169-160 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt9}{2.1} & & = \frac{-13+\sqrt9}{2.1} \\
& = \frac{-16}{2} & & = \frac{-10}{2} \\
& = -8 & & = -5 \\ \\ V &= \Big\{ -8 ; -5 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{4}x^2-\frac{1}{2}x+\frac{16}{9}=0\\
\Leftrightarrow \color{red}{36.} \left(\frac{1}{4}x^2-\frac{1}{2}x+\frac{16}{9}\right)=0 \color{red}{.36} \\
\Leftrightarrow 9x^2-18x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-18x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-18)^2-4.9.64 & &\\
& = 324-2304 & & \\
& = -1980 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(-(14-10x)=-x^2-(4-7x) \\
\Leftrightarrow -14+10x=-x^2-4+7x \\
\Leftrightarrow x^2+3x-10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-10=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.1.(-10) & &\\
& = 9+40 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt49}{2.1} & & = \frac{-3+\sqrt49}{2.1} \\
& = \frac{-10}{2} & & = \frac{4}{2} \\
& = -5 & & = 2 \\ \\ V &= \Big\{ -5 ; 2 \Big\} & &\end{align} \\ -----------------\)
- \((4x+4)(-5x-4)-x(-92x-57)=-18\\
\Leftrightarrow -20x^2-16x-20x-16 +92x^2+57x+18=0 \\
\Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.72.2 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\
& = \frac{-32}{144} & & = \frac{-18}{144} \\
& = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \((x-5)(5x-2)-x(4x+24)=-54\\
\Leftrightarrow 5x^2-2x-25x+10 -4x^2-24x+54=0 \\
\Leftrightarrow x^2-16x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-16)^2-4.1.64 & &\\
& = 256-256 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-16)}{2.1} & & \\
& = 8 & & \\V &= \Big\{ 8 \Big\} & &\end{align} \\ -----------------\)
- \(x(4x+1)=-(x+1) \\
\Leftrightarrow 4x^2+x=-x-1 \\
\Leftrightarrow 4x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.4.1 & &\\
& = 4-16 & & \\
& = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\frac{5}{3}x=-3x^2+\frac{4}{3} \\
\Leftrightarrow 3x^2+\frac{5}{3}x-\frac{4}{3}=0 \\
\Leftrightarrow \color{red}{3.} \left(3x^2+\frac{5}{3}x-\frac{4}{3}\right)=0 \color{red}{.3} \\
\Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.9.(-4) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\
& = \frac{-18}{18} & & = \frac{8}{18} \\
& = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
- \(5x^2-(6x+36)=x(x-13) \\
\Leftrightarrow 5x^2-6x-36=x^2-13x \\
\Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.4.(-36) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\
& = \frac{-32}{8} & & = \frac{18}{8} \\
& = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
- \(-(14-45x)=-9x^2-(39-15x) \\
\Leftrightarrow -14+45x=-9x^2-39+15x \\
\Leftrightarrow 9x^2+30x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+30x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (30)^2-4.9.25 & &\\
& = 900-900 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-30}{2.9} & & \\
& = -\frac{5}{3} & & \\V &= \Big\{ -\frac{5}{3} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{2}x^2+\frac{19}{2}x+45=0\\
\Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2+\frac{19}{2}x+45\right)=0 \color{red}{.2} \\
\Leftrightarrow x^2+19x+90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+19x+90=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (19)^2-4.1.90 & &\\
& = 361-360 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-19-\sqrt1}{2.1} & & = \frac{-19+\sqrt1}{2.1} \\
& = \frac{-20}{2} & & = \frac{-18}{2} \\
& = -10 & & = -9 \\ \\ V &= \Big\{ -10 ; -9 \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{1}{2}x=-\frac{1}{8}x^2-\frac{1}{2} \\
\Leftrightarrow \frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}=0 \\
\Leftrightarrow \color{red}{8.} \left(\frac{1}{8}x^2-\frac{1}{2}x+\frac{1}{2}\right)=0 \color{red}{.8} \\
\Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-64)^2-4.16.64 & &\\
& = 4096-4096 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-64)}{2.16} & & \\
& = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
- \(x(16x+9)=-(x+1) \\
\Leftrightarrow 16x^2+9x=-x-1 \\
\Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (10)^2-4.16.1 & &\\
& = 100-64 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\
& = \frac{-16}{32} & & = \frac{-4}{32} \\
& = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)