Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(-(7-38x)=-24x^2-(13-13x)\)
- \((x+1)(3x+3)-x(2x-1)=47\)
- \(-(10-39x)=-2x^2-(82-14x)\)
- \((-3x+2)(3x+4)-x(-10x+18)=-113\)
- \(-(13+x)=-x^2-(62-13x)\)
- \((x-3)(4x+4)-x(-12x-64)=-61\)
- \(\frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}=0\)
- \(\frac{1}{2}x^2-5x-\frac{11}{2}=0\)
- \((-x-5)(2x-4)-x(-5x+19)=32\)
- \(2x^2-(13x+27)=x(x-19)\)
- \(x(x+26)=5(x-22)\)
- \(\frac{17}{10}x=-\frac{1}{5}x^2-\frac{4}{5}\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(-(7-38x)=-24x^2-(13-13x) \\
\Leftrightarrow -7+38x=-24x^2-13+13x \\
\Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.24.6 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\
& = \frac{-32}{48} & & = \frac{-18}{48} \\
& = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
- \((x+1)(3x+3)-x(2x-1)=47\\
\Leftrightarrow 3x^2+3x+3x+3 -2x^2+x-47=0 \\
\Leftrightarrow x^2+7x-44=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+7x-44=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.1.(-44) & &\\
& = 49+176 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt225}{2.1} & & = \frac{-7+\sqrt225}{2.1} \\
& = \frac{-22}{2} & & = \frac{8}{2} \\
& = -11 & & = 4 \\ \\ V &= \Big\{ -11 ; 4 \Big\} & &\end{align} \\ -----------------\)
- \(-(10-39x)=-2x^2-(82-14x) \\
\Leftrightarrow -10+39x=-2x^2-82+14x \\
\Leftrightarrow 2x^2+25x+72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+25x+72=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.2.72 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.2} & & = \frac{-25+\sqrt49}{2.2} \\
& = \frac{-32}{4} & & = \frac{-18}{4} \\
& = -8 & & = \frac{-9}{2} \\ \\ V &= \Big\{ -8 ; \frac{-9}{2} \Big\} & &\end{align} \\ -----------------\)
- \((-3x+2)(3x+4)-x(-10x+18)=-113\\
\Leftrightarrow -9x^2-12x+6x+8 +10x^2-18x+113=0 \\
\Leftrightarrow x^2-22x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-22x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-22)^2-4.1.121 & &\\
& = 484-484 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-22)}{2.1} & & \\
& = 11 & & \\V &= \Big\{ 11 \Big\} & &\end{align} \\ -----------------\)
- \(-(13+x)=-x^2-(62-13x) \\
\Leftrightarrow -13-x=-x^2-62+13x \\
\Leftrightarrow x^2-14x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-14x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-14)^2-4.1.49 & &\\
& = 196-196 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-14)}{2.1} & & \\
& = 7 & & \\V &= \Big\{ 7 \Big\} & &\end{align} \\ -----------------\)
- \((x-3)(4x+4)-x(-12x-64)=-61\\
\Leftrightarrow 4x^2+4x-12x-12 +12x^2+64x+61=0 \\
\Leftrightarrow 16x^2+56x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+56x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (56)^2-4.16.49 & &\\
& = 3136-3136 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-56}{2.16} & & \\
& = -\frac{7}{4} & & \\V &= \Big\{ -\frac{7}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}=0\\
\Leftrightarrow \color{red}{18.} \left(\frac{1}{3}x^2+\frac{5}{18}x-\frac{1}{3}\right)=0 \color{red}{.18} \\
\Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.6.(-6) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\
& = \frac{-18}{12} & & = \frac{8}{12} \\
& = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{2}x^2-5x-\frac{11}{2}=0\\
\Leftrightarrow \color{red}{2.} \left(\frac{1}{2}x^2-5x-\frac{11}{2}\right)=0 \color{red}{.2} \\
\Leftrightarrow x^2-10x-11=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x-11=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-10)^2-4.1.(-11) & &\\
& = 100+44 & & \\
& = 144 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-10)-\sqrt144}{2.1} & & = \frac{-(-10)+\sqrt144}{2.1} \\
& = \frac{-2}{2} & & = \frac{22}{2} \\
& = -1 & & = 11 \\ \\ V &= \Big\{ -1 ; 11 \Big\} & &\end{align} \\ -----------------\)
- \((-x-5)(2x-4)-x(-5x+19)=32\\
\Leftrightarrow -2x^2+4x-10x+20 +5x^2-19x-32=0 \\
\Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.3.(-12) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\
& = \frac{-18}{6} & & = \frac{8}{6} \\
& = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(13x+27)=x(x-19) \\
\Leftrightarrow 2x^2-13x-27=x^2-19x \\
\Leftrightarrow x^2+6x-27=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+6x-27=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.1.(-27) & &\\
& = 36+108 & & \\
& = 144 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-6-\sqrt144}{2.1} & & = \frac{-6+\sqrt144}{2.1} \\
& = \frac{-18}{2} & & = \frac{6}{2} \\
& = -9 & & = 3 \\ \\ V &= \Big\{ -9 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+26)=5(x-22) \\
\Leftrightarrow x^2+26x=5x-110 \\
\Leftrightarrow x^2+21x+110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+21x+110=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (21)^2-4.1.110 & &\\
& = 441-440 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-21-\sqrt1}{2.1} & & = \frac{-21+\sqrt1}{2.1} \\
& = \frac{-22}{2} & & = \frac{-20}{2} \\
& = -11 & & = -10 \\ \\ V &= \Big\{ -11 ; -10 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{17}{10}x=-\frac{1}{5}x^2-\frac{4}{5} \\
\Leftrightarrow \frac{1}{5}x^2+\frac{17}{10}x+\frac{4}{5}=0 \\
\Leftrightarrow \color{red}{10.} \left(\frac{1}{5}x^2+\frac{17}{10}x+\frac{4}{5}\right)=0 \color{red}{.10} \\
\Leftrightarrow 2x^2+17x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+17x+8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.2.8 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.2} & & = \frac{-17+\sqrt225}{2.2} \\
& = \frac{-32}{4} & & = \frac{-2}{4} \\
& = -8 & & = \frac{-1}{2} \\ \\ V &= \Big\{ -8 ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)