Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(\frac{25}{12}x=-4x^2-\frac{1}{4}\)
- \(x(4x+50)=10(x-10)\)
- \((5x-4)(-2x-3)-x(-14x-6)=13\)
- \(2x^2-(13x-16)=x(x-5)\)
- \(x(x+32)=8(x-18)\)
- \(10x^2-(15x-36)=x(x+21)\)
- \(2x^2+\frac{17}{4}x+\frac{1}{2}=0\)
- \(4x^2-(13x-24)=3x(x-9)\)
- \((-2x-4)(-3x+2)-x(-10x-29)=-9\)
- \(x(x+21)=3(x-27)\)
- \(\frac{1}{2}x=-\frac{1}{5}x^2+\frac{9}{5}\)
- \(-(6-28x)=-18x^2-(14-3x)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\frac{25}{12}x=-4x^2-\frac{1}{4} \\
\Leftrightarrow 4x^2+\frac{25}{12}x+\frac{1}{4}=0 \\
\Leftrightarrow \color{red}{12.} \left(4x^2+\frac{25}{12}x+\frac{1}{4}\right)=0 \color{red}{.12} \\
\Leftrightarrow 48x^2+25x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{48x^2+25x+3=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.48.3 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.48} & & = \frac{-25+\sqrt49}{2.48} \\
& = \frac{-32}{96} & & = \frac{-18}{96} \\
& = \frac{-1}{3} & & = \frac{-3}{16} \\ \\ V &= \Big\{ \frac{-1}{3} ; \frac{-3}{16} \Big\} & &\end{align} \\ -----------------\)
- \(x(4x+50)=10(x-10) \\
\Leftrightarrow 4x^2+50x=10x-100 \\
\Leftrightarrow 4x^2+40x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+40x+100=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (40)^2-4.4.100 & &\\
& = 1600-1600 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-40}{2.4} & & \\
& = -5 & & \\V &= \Big\{ -5 \Big\} & &\end{align} \\ -----------------\)
- \((5x-4)(-2x-3)-x(-14x-6)=13\\
\Leftrightarrow -10x^2-15x+8x+12 +14x^2+6x-13=0 \\
\Leftrightarrow 4x^2+3x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+3x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.4.(-1) & &\\
& = 9+16 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt25}{2.4} & & = \frac{-3+\sqrt25}{2.4} \\
& = \frac{-8}{8} & & = \frac{2}{8} \\
& = -1 & & = \frac{1}{4} \\ \\ V &= \Big\{ -1 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(13x-16)=x(x-5) \\
\Leftrightarrow 2x^2-13x+16=x^2-5x \\
\Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-8)^2-4.1.16 & &\\
& = 64-64 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-8)}{2.1} & & \\
& = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+32)=8(x-18) \\
\Leftrightarrow x^2+32x=8x-144 \\
\Leftrightarrow x^2+24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+24x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (24)^2-4.1.144 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-24}{2.1} & & \\
& = -12 & & \\V &= \Big\{ -12 \Big\} & &\end{align} \\ -----------------\)
- \(10x^2-(15x-36)=x(x+21) \\
\Leftrightarrow 10x^2-15x+36=x^2+21x \\
\Leftrightarrow 9x^2-36x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-36x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-36)^2-4.9.36 & &\\
& = 1296-1296 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-36)}{2.9} & & \\
& = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2+\frac{17}{4}x+\frac{1}{2}=0\\
\Leftrightarrow \color{red}{4.} \left(2x^2+\frac{17}{4}x+\frac{1}{2}\right)=0 \color{red}{.4} \\
\Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.8.2 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\
& = \frac{-32}{16} & & = \frac{-2}{16} \\
& = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(4x^2-(13x-24)=3x(x-9) \\
\Leftrightarrow 4x^2-13x+24=3x^2-27x \\
\Leftrightarrow x^2+14x+24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+24=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (14)^2-4.1.24 & &\\
& = 196-96 & & \\
& = 100 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-14-\sqrt100}{2.1} & & = \frac{-14+\sqrt100}{2.1} \\
& = \frac{-24}{2} & & = \frac{-4}{2} \\
& = -12 & & = -2 \\ \\ V &= \Big\{ -12 ; -2 \Big\} & &\end{align} \\ -----------------\)
- \((-2x-4)(-3x+2)-x(-10x-29)=-9\\
\Leftrightarrow 6x^2-4x+12x-8 +10x^2+29x+9=0 \\
\Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.16.1 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\
& = \frac{-32}{32} & & = \frac{-2}{32} \\
& = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
- \(x(x+21)=3(x-27) \\
\Leftrightarrow x^2+21x=3x-81 \\
\Leftrightarrow x^2+18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+18x+81=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (18)^2-4.1.81 & &\\
& = 324-324 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-18}{2.1} & & \\
& = -9 & & \\V &= \Big\{ -9 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{2}x=-\frac{1}{5}x^2+\frac{9}{5} \\
\Leftrightarrow \frac{1}{5}x^2+\frac{1}{2}x-\frac{9}{5}=0 \\
\Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{1}{2}x-\frac{9}{5}\right)=0 \color{red}{.10} \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(-(6-28x)=-18x^2-(14-3x) \\
\Leftrightarrow -6+28x=-18x^2-14+3x \\
\Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.18.8 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\
& = \frac{-32}{36} & & = \frac{-18}{36} \\
& = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)