Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(2x^2-(10x-72)=x(x+7)\)
- \(\frac{13}{4}x=-x^2-\frac{9}{4}\)
- \(21x^2-(20x-2)=3x(x-11)\)
- \(-(9-19x)=-x^2-(-21-12x)\)
- \(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}=0\)
- \(\frac{1}{4}x^2+\frac{17}{4}x+\frac{35}{2}=0\)
- \(24x^2-(16x-1)=8x(x-3)\)
- \(\frac{1}{4}x^2+\frac{3}{4}x-\frac{35}{2}=0\)
- \((3x+3)(3x-3)-x(5x-34)=-25\)
- \(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}=0\)
- \(\frac{1}{2}x^2-\frac{9}{2}x+4=0\)
- \(17x^2-(7x+1)=x(x-22)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(2x^2-(10x-72)=x(x+7) \\
\Leftrightarrow 2x^2-10x+72=x^2+7x \\
\Leftrightarrow x^2-17x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-17x+72=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-17)^2-4.1.72 & &\\
& = 289-288 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-17)-\sqrt1}{2.1} & & = \frac{-(-17)+\sqrt1}{2.1} \\
& = \frac{16}{2} & & = \frac{18}{2} \\
& = 8 & & = 9 \\ \\ V &= \Big\{ 8 ; 9 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{13}{4}x=-x^2-\frac{9}{4} \\
\Leftrightarrow x^2+\frac{13}{4}x+\frac{9}{4}=0 \\
\Leftrightarrow \color{red}{4.} \left(x^2+\frac{13}{4}x+\frac{9}{4}\right)=0 \color{red}{.4} \\
\Leftrightarrow 4x^2+13x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+13x+9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.4.9 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.4} & & = \frac{-13+\sqrt25}{2.4} \\
& = \frac{-18}{8} & & = \frac{-8}{8} \\
& = \frac{-9}{4} & & = -1 \\ \\ V &= \Big\{ \frac{-9}{4} ; -1 \Big\} & &\end{align} \\ -----------------\)
- \(21x^2-(20x-2)=3x(x-11) \\
\Leftrightarrow 21x^2-20x+2=3x^2-33x \\
\Leftrightarrow 18x^2+13x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+13x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.18.2 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.18} & & = \frac{-13+\sqrt25}{2.18} \\
& = \frac{-18}{36} & & = \frac{-8}{36} \\
& = \frac{-1}{2} & & = \frac{-2}{9} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-2}{9} \Big\} & &\end{align} \\ -----------------\)
- \(-(9-19x)=-x^2-(-21-12x) \\
\Leftrightarrow -9+19x=-x^2+21+12x \\
\Leftrightarrow x^2+7x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.1.(-30) & &\\
& = 49+120 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt169}{2.1} & & = \frac{-7+\sqrt169}{2.1} \\
& = \frac{-20}{2} & & = \frac{6}{2} \\
& = -10 & & = 3 \\ \\ V &= \Big\{ -10 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}=0\\
\Leftrightarrow \color{red}{9.} \left(\frac{4}{3}x^2+\frac{5}{9}x-\frac{1}{3}\right)=0 \color{red}{.9} \\
\Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.12.(-3) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\
& = \frac{-18}{24} & & = \frac{8}{24} \\
& = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{4}x^2+\frac{17}{4}x+\frac{35}{2}=0\\
\Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{17}{4}x+\frac{35}{2}\right)=0 \color{red}{.4} \\
\Leftrightarrow x^2+17x+70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+17x+70=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.1.70 & &\\
& = 289-280 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt9}{2.1} & & = \frac{-17+\sqrt9}{2.1} \\
& = \frac{-20}{2} & & = \frac{-14}{2} \\
& = -10 & & = -7 \\ \\ V &= \Big\{ -10 ; -7 \Big\} & &\end{align} \\ -----------------\)
- \(24x^2-(16x-1)=8x(x-3) \\
\Leftrightarrow 24x^2-16x+1=8x^2-24x \\
\Leftrightarrow 16x^2+8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (8)^2-4.16.1 & &\\
& = 64-64 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-8}{2.16} & & \\
& = -\frac{1}{4} & & \\V &= \Big\{ -\frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{4}x^2+\frac{3}{4}x-\frac{35}{2}=0\\
\Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2+\frac{3}{4}x-\frac{35}{2}\right)=0 \color{red}{.4} \\
\Leftrightarrow x^2+3x-70=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-70=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.1.(-70) & &\\
& = 9+280 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt289}{2.1} & & = \frac{-3+\sqrt289}{2.1} \\
& = \frac{-20}{2} & & = \frac{14}{2} \\
& = -10 & & = 7 \\ \\ V &= \Big\{ -10 ; 7 \Big\} & &\end{align} \\ -----------------\)
- \((3x+3)(3x-3)-x(5x-34)=-25\\
\Leftrightarrow 9x^2-9x+9x-9 -5x^2+34x+25=0 \\
\Leftrightarrow 4x^2+16x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+16x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (16)^2-4.4.16 & &\\
& = 256-256 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-16}{2.4} & & \\
& = -2 & & \\V &= \Big\{ -2 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}=0\\
\Leftrightarrow \color{red}{10.} \left(\frac{36}{5}x^2+\frac{5}{2}x+\frac{1}{5}\right)=0 \color{red}{.10} \\
\Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.72.2 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\
& = \frac{-32}{144} & & = \frac{-18}{144} \\
& = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{2}x^2-\frac{9}{2}x+4=0\\
\Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-\frac{9}{2}x+4\right)=0 \color{red}{.2} \\
\Leftrightarrow x^2-9x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-9x+8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-9)^2-4.1.8 & &\\
& = 81-32 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-9)-\sqrt49}{2.1} & & = \frac{-(-9)+\sqrt49}{2.1} \\
& = \frac{2}{2} & & = \frac{16}{2} \\
& = 1 & & = 8 \\ \\ V &= \Big\{ 1 ; 8 \Big\} & &\end{align} \\ -----------------\)
- \(17x^2-(7x+1)=x(x-22) \\
\Leftrightarrow 17x^2-7x-1=x^2-22x \\
\Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.16.(-1) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\
& = \frac{-32}{32} & & = \frac{2}{32} \\
& = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)