Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(\frac{4}{3}x^2+\frac{7}{6}x-3=0\)
- \((4x+1)(5x-3)-x(19x-16)=87\)
- \(-(3+13x)=-x^2-(87-6x)\)
- \(\frac{1}{3}x^2-\frac{13}{3}x+10=0\)
- \(-(13-16x)=-x^2-(22-10x)\)
- \(x(16x+1)=5(x-20)\)
- \(\frac{25}{16}x=-\frac{9}{4}x^2-\frac{1}{4}\)
- \(2x=-\frac{3}{7}x^2-\frac{7}{3}\)
- \(x(4x-149)=-121(x+1)\)
- \((-3x+1)(4x-2)-x(-28x-11)=-1\)
- \(x(x+17)=2(x-27)\)
- \(10x^2-(8x-49)=x(x+34)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\frac{4}{3}x^2+\frac{7}{6}x-3=0\\
\Leftrightarrow \color{red}{6.} \left(\frac{4}{3}x^2+\frac{7}{6}x-3\right)=0 \color{red}{.6} \\
\Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.8.(-18) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\
& = \frac{-32}{16} & & = \frac{18}{16} \\
& = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
- \((4x+1)(5x-3)-x(19x-16)=87\\
\Leftrightarrow 20x^2-12x+5x-3 -19x^2+16x-87=0 \\
\Leftrightarrow x^2+x-90=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-90=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (1)^2-4.1.(-90) & &\\
& = 1+360 & & \\
& = 361 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-1-\sqrt361}{2.1} & & = \frac{-1+\sqrt361}{2.1} \\
& = \frac{-20}{2} & & = \frac{18}{2} \\
& = -10 & & = 9 \\ \\ V &= \Big\{ -10 ; 9 \Big\} & &\end{align} \\ -----------------\)
- \(-(3+13x)=-x^2-(87-6x) \\
\Leftrightarrow -3-13x=-x^2-87+6x \\
\Leftrightarrow x^2-19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+84=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-19)^2-4.1.84 & &\\
& = 361-336 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-19)-\sqrt25}{2.1} & & = \frac{-(-19)+\sqrt25}{2.1} \\
& = \frac{14}{2} & & = \frac{24}{2} \\
& = 7 & & = 12 \\ \\ V &= \Big\{ 7 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2-\frac{13}{3}x+10=0\\
\Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{13}{3}x+10\right)=0 \color{red}{.3} \\
\Leftrightarrow x^2-13x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-13)^2-4.1.30 & &\\
& = 169-120 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-13)-\sqrt49}{2.1} & & = \frac{-(-13)+\sqrt49}{2.1} \\
& = \frac{6}{2} & & = \frac{20}{2} \\
& = 3 & & = 10 \\ \\ V &= \Big\{ 3 ; 10 \Big\} & &\end{align} \\ -----------------\)
- \(-(13-16x)=-x^2-(22-10x) \\
\Leftrightarrow -13+16x=-x^2-22+10x \\
\Leftrightarrow x^2+6x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x+9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.1.9 & &\\
& = 36-36 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-6}{2.1} & & \\
& = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
- \(x(16x+1)=5(x-20) \\
\Leftrightarrow 16x^2+x=5x-100 \\
\Leftrightarrow 16x^2-4x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-4x+100=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.16.100 & &\\
& = 16-6400 & & \\
& = -6384 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\frac{25}{16}x=-\frac{9}{4}x^2-\frac{1}{4} \\
\Leftrightarrow \frac{9}{4}x^2+\frac{25}{16}x+\frac{1}{4}=0 \\
\Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{25}{16}x+\frac{1}{4}\right)=0 \color{red}{.16} \\
\Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.36.4 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\
& = \frac{-32}{72} & & = \frac{-18}{72} \\
& = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(2x=-\frac{3}{7}x^2-\frac{7}{3} \\
\Leftrightarrow \frac{3}{7}x^2+2x+\frac{7}{3}=0 \\
\Leftrightarrow \color{red}{21.} \left(\frac{3}{7}x^2+2x+\frac{7}{3}\right)=0 \color{red}{.21} \\
\Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (42)^2-4.9.49 & &\\
& = 1764-1764 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-42}{2.9} & & \\
& = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
- \(x(4x-149)=-121(x+1) \\
\Leftrightarrow 4x^2-149x=-121x-121 \\
\Leftrightarrow 4x^2-28x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-28x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-28)^2-4.4.121 & &\\
& = 784-1936 & & \\
& = -1152 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \((-3x+1)(4x-2)-x(-28x-11)=-1\\
\Leftrightarrow -12x^2+6x+4x-2 +28x^2+11x+1=0 \\
\Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.16.(-1) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\
& = \frac{-32}{32} & & = \frac{2}{32} \\
& = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
- \(x(x+17)=2(x-27) \\
\Leftrightarrow x^2+17x=2x-54 \\
\Leftrightarrow x^2+15x+54=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+15x+54=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.1.54 & &\\
& = 225-216 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt9}{2.1} & & = \frac{-15+\sqrt9}{2.1} \\
& = \frac{-18}{2} & & = \frac{-12}{2} \\
& = -9 & & = -6 \\ \\ V &= \Big\{ -9 ; -6 \Big\} & &\end{align} \\ -----------------\)
- \(10x^2-(8x-49)=x(x+34) \\
\Leftrightarrow 10x^2-8x+49=x^2+34x \\
\Leftrightarrow 9x^2-42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-42x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-42)^2-4.9.49 & &\\
& = 1764-1764 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-42)}{2.9} & & \\
& = \frac{7}{3} & & \\V &= \Big\{ \frac{7}{3} \Big\} & &\end{align} \\ -----------------\)