Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(-(10-41x)=-18x^2-(18-16x)\)
- \(-\frac{7}{3}x=-\frac{1}{3}x^2+\frac{8}{3}\)
- \(x(x+1)=-(x+1)\)
- \(5x^2-(17x+12)=2x(x-11)\)
- \((x+5)(x+3)-x(-3x+13)=24\)
- \((-4x-3)(-4x+2)-x(12x-12)=-7\)
- \(39x^2-(14x+4)=3x(x-7)\)
- \(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}=0\)
- \(-(3-34x)=-16x^2-(39-18x)\)
- \(17x^2-(8x-121)=x(x-20)\)
- \(10x^2-(13x-33)=9x(x-3)\)
- \(x(x+19)=24(x+1)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(-(10-41x)=-18x^2-(18-16x) \\
\Leftrightarrow -10+41x=-18x^2-18+16x \\
\Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.18.8 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\
& = \frac{-32}{36} & & = \frac{-18}{36} \\
& = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{7}{3}x=-\frac{1}{3}x^2+\frac{8}{3} \\
\Leftrightarrow \frac{1}{3}x^2-\frac{7}{3}x-\frac{8}{3}=0 \\
\Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{7}{3}x-\frac{8}{3}\right)=0 \color{red}{.3} \\
\Leftrightarrow x^2-7x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-7x-8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-7)^2-4.1.(-8) & &\\
& = 49+32 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-7)-\sqrt81}{2.1} & & = \frac{-(-7)+\sqrt81}{2.1} \\
& = \frac{-2}{2} & & = \frac{16}{2} \\
& = -1 & & = 8 \\ \\ V &= \Big\{ -1 ; 8 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+1)=-(x+1) \\
\Leftrightarrow x^2+x=-x-1 \\
\Leftrightarrow x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.1 & &\\
& = 4-4 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-2}{2.1} & & \\
& = -1 & & \\V &= \Big\{ -1 \Big\} & &\end{align} \\ -----------------\)
- \(5x^2-(17x+12)=2x(x-11) \\
\Leftrightarrow 5x^2-17x-12=2x^2-22x \\
\Leftrightarrow 3x^2+5x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{3x^2+5x-12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.3.(-12) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.3} & & = \frac{-5+\sqrt169}{2.3} \\
& = \frac{-18}{6} & & = \frac{8}{6} \\
& = -3 & & = \frac{4}{3} \\ \\ V &= \Big\{ -3 ; \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
- \((x+5)(x+3)-x(-3x+13)=24\\
\Leftrightarrow x^2+3x+5x+15 +3x^2-13x-24=0 \\
\Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.4.(-9) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\
& = \frac{-18}{8} & & = \frac{8}{8} \\
& = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
- \((-4x-3)(-4x+2)-x(12x-12)=-7\\
\Leftrightarrow 16x^2-8x+12x-6 -12x^2+12x+7=0 \\
\Leftrightarrow 4x^2-2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.4.1 & &\\
& = 4-16 & & \\
& = -12 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(39x^2-(14x+4)=3x(x-7) \\
\Leftrightarrow 39x^2-14x-4=3x^2-21x \\
\Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.36.(-4) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\
& = \frac{-32}{72} & & = \frac{18}{72} \\
& = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}=0\\
\Leftrightarrow \color{red}{36.} \left(\frac{1}{3}x^2+\frac{7}{36}x-\frac{1}{3}\right)=0 \color{red}{.36} \\
\Leftrightarrow 12x^2+7x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+7x-12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.12.(-12) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.12} & & = \frac{-7+\sqrt625}{2.12} \\
& = \frac{-32}{24} & & = \frac{18}{24} \\
& = \frac{-4}{3} & & = \frac{3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(-(3-34x)=-16x^2-(39-18x) \\
\Leftrightarrow -3+34x=-16x^2-39+18x \\
\Leftrightarrow 16x^2+16x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+16x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (16)^2-4.16.36 & &\\
& = 256-2304 & & \\
& = -2048 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(17x^2-(8x-121)=x(x-20) \\
\Leftrightarrow 17x^2-8x+121=x^2-20x \\
\Leftrightarrow 16x^2+12x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+12x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (12)^2-4.16.121 & &\\
& = 144-7744 & & \\
& = -7600 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(10x^2-(13x-33)=9x(x-3) \\
\Leftrightarrow 10x^2-13x+33=9x^2-27x \\
\Leftrightarrow x^2+14x+33=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+14x+33=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (14)^2-4.1.33 & &\\
& = 196-132 & & \\
& = 64 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-14-\sqrt64}{2.1} & & = \frac{-14+\sqrt64}{2.1} \\
& = \frac{-22}{2} & & = \frac{-6}{2} \\
& = -11 & & = -3 \\ \\ V &= \Big\{ -11 ; -3 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+19)=24(x+1) \\
\Leftrightarrow x^2+19x=24x+24 \\
\Leftrightarrow x^2-5x-24=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-24=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-5)^2-4.1.(-24) & &\\
& = 25+96 & & \\
& = 121 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-5)-\sqrt121}{2.1} & & = \frac{-(-5)+\sqrt121}{2.1} \\
& = \frac{-6}{2} & & = \frac{16}{2} \\
& = -3 & & = 8 \\ \\ V &= \Big\{ -3 ; 8 \Big\} & &\end{align} \\ -----------------\)