Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(-(7+2x)=-9x^2-(8-4x)\)
- \(\frac{7}{8}x=-9x^2+\frac{1}{4}\)
- \(x(x-9)=2(x-14)\)
- \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}=0\)
- \(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}=0\)
- \(-(2+3x)=-x^2-(-34-6x)\)
- \(x(16x+16)=-(x+1)\)
- \((4x+2)(-3x-5)-x(-30x-43)=-12\)
- \(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}=0\)
- \(x(9x-22)=2(x-8)\)
- \(x(8x+17)=2(x+1)\)
- \(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}=0\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(-(7+2x)=-9x^2-(8-4x) \\
\Leftrightarrow -7-2x=-9x^2-8+4x \\
\Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-6)^2-4.9.1 & &\\
& = 36-36 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-6)}{2.9} & & \\
& = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{7}{8}x=-9x^2+\frac{1}{4} \\
\Leftrightarrow 9x^2+\frac{7}{8}x-\frac{1}{4}=0 \\
\Leftrightarrow \color{red}{8.} \left(9x^2+\frac{7}{8}x-\frac{1}{4}\right)=0 \color{red}{.8} \\
\Leftrightarrow 72x^2+7x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+7x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.72.(-2) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.72} & & = \frac{-7+\sqrt625}{2.72} \\
& = \frac{-32}{144} & & = \frac{18}{144} \\
& = \frac{-2}{9} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(x(x-9)=2(x-14) \\
\Leftrightarrow x^2-9x=2x-28 \\
\Leftrightarrow x^2-11x+28=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+28=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-11)^2-4.1.28 & &\\
& = 121-112 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-11)-\sqrt9}{2.1} & & = \frac{-(-11)+\sqrt9}{2.1} \\
& = \frac{8}{2} & & = \frac{14}{2} \\
& = 4 & & = 7 \\ \\ V &= \Big\{ 4 ; 7 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}=0\\
\Leftrightarrow \color{red}{4.} \left(\frac{1}{2}x^2+\frac{5}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\
\Leftrightarrow 2x^2+5x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.2 & &\\
& = 25-16 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt9}{2.2} & & = \frac{-5+\sqrt9}{2.2} \\
& = \frac{-8}{4} & & = \frac{-2}{4} \\
& = -2 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -2 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}=0\\
\Leftrightarrow \color{red}{5.} \left(\frac{4}{5}x^2+\frac{13}{5}x+\frac{9}{5}\right)=0 \color{red}{.5} \\
\Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.4.9 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\
& = \frac{-18}{8} & & = \frac{-8}{8} \\
& = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
- \(-(2+3x)=-x^2-(-34-6x) \\
\Leftrightarrow -2-3x=-x^2+34+6x \\
\Leftrightarrow x^2-9x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x-36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-9)^2-4.1.(-36) & &\\
& = 81+144 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-9)-\sqrt225}{2.1} & & = \frac{-(-9)+\sqrt225}{2.1} \\
& = \frac{-6}{2} & & = \frac{24}{2} \\
& = -3 & & = 12 \\ \\ V &= \Big\{ -3 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \(x(16x+16)=-(x+1) \\
\Leftrightarrow 16x^2+16x=-x-1 \\
\Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.16.1 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\
& = \frac{-32}{32} & & = \frac{-2}{32} \\
& = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
- \((4x+2)(-3x-5)-x(-30x-43)=-12\\
\Leftrightarrow -12x^2-20x-6x-10 +30x^2+43x+12=0 \\
\Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.18.2 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\
& = \frac{-18}{36} & & = \frac{-8}{36} \\
& = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}=0\\
\Leftrightarrow \color{red}{15.} \left(\frac{3}{5}x^2-\frac{8}{5}x+\frac{5}{3}\right)=0 \color{red}{.15} \\
\Leftrightarrow 9x^2-24x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.9.25 & &\\
& = 576-900 & & \\
& = -324 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x(9x-22)=2(x-8) \\
\Leftrightarrow 9x^2-22x=2x-16 \\
\Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.9.16 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.9} & & \\
& = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
- \(x(8x+17)=2(x+1) \\
\Leftrightarrow 8x^2+17x=2x+2 \\
\Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.8.(-2) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\
& = \frac{-32}{16} & & = \frac{2}{16} \\
& = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}=0\\
\Leftrightarrow \color{red}{16.} \left(\frac{1}{4}x^2+\frac{15}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\
\Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.4.(-4) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\
& = \frac{-32}{8} & & = \frac{2}{8} \\
& = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)