Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(\frac{5}{4}x=-\frac{9}{4}x^2+1\)
- \(-(13-14x)=-x^2-(113-18x)\)
- \(2x^2-(17x-10)=x(x-6)\)
- \(14x^2-(2x-144)=13x(x-2)\)
- \(-(12+15x)=-x^2-(92-3x)\)
- \((-2x-3)(-3x+1)-x(2x-20)=1\)
- \((2x+1)(5x+2)-x(9x+2)=-14\)
- \(\frac{1}{2}x^2+\frac{25}{24}x+\frac{1}{2}=0\)
- \(x(x+13)=15(x+1)\)
- \(2x^2+\frac{13}{6}x+\frac{1}{2}=0\)
- \(\frac{1}{3}x^2+\frac{7}{12}x-3=0\)
- \(-(5-21x)=-x^2-(-1-20x)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(\frac{5}{4}x=-\frac{9}{4}x^2+1 \\
\Leftrightarrow \frac{9}{4}x^2+\frac{5}{4}x-1=0 \\
\Leftrightarrow \color{red}{4.} \left(\frac{9}{4}x^2+\frac{5}{4}x-1\right)=0 \color{red}{.4} \\
\Leftrightarrow 9x^2+5x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+5x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.9.(-4) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.9} & & = \frac{-5+\sqrt169}{2.9} \\
& = \frac{-18}{18} & & = \frac{8}{18} \\
& = -1 & & = \frac{4}{9} \\ \\ V &= \Big\{ -1 ; \frac{4}{9} \Big\} & &\end{align} \\ -----------------\)
- \(-(13-14x)=-x^2-(113-18x) \\
\Leftrightarrow -13+14x=-x^2-113+18x \\
\Leftrightarrow x^2-4x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-4x+100=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.1.100 & &\\
& = 16-400 & & \\
& = -384 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(2x^2-(17x-10)=x(x-6) \\
\Leftrightarrow 2x^2-17x+10=x^2-6x \\
\Leftrightarrow x^2-11x+10=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+10=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-11)^2-4.1.10 & &\\
& = 121-40 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-11)-\sqrt81}{2.1} & & = \frac{-(-11)+\sqrt81}{2.1} \\
& = \frac{2}{2} & & = \frac{20}{2} \\
& = 1 & & = 10 \\ \\ V &= \Big\{ 1 ; 10 \Big\} & &\end{align} \\ -----------------\)
- \(14x^2-(2x-144)=13x(x-2) \\
\Leftrightarrow 14x^2-2x+144=13x^2-26x \\
\Leftrightarrow x^2+24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+24x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (24)^2-4.1.144 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-24}{2.1} & & \\
& = -12 & & \\V &= \Big\{ -12 \Big\} & &\end{align} \\ -----------------\)
- \(-(12+15x)=-x^2-(92-3x) \\
\Leftrightarrow -12-15x=-x^2-92+3x \\
\Leftrightarrow x^2-18x+80=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+80=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-18)^2-4.1.80 & &\\
& = 324-320 & & \\
& = 4 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-18)-\sqrt4}{2.1} & & = \frac{-(-18)+\sqrt4}{2.1} \\
& = \frac{16}{2} & & = \frac{20}{2} \\
& = 8 & & = 10 \\ \\ V &= \Big\{ 8 ; 10 \Big\} & &\end{align} \\ -----------------\)
- \((-2x-3)(-3x+1)-x(2x-20)=1\\
\Leftrightarrow 6x^2-2x+9x-3 -2x^2+20x-1=0 \\
\Leftrightarrow 4x^2+15x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+15x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.4.(-4) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.4} & & = \frac{-15+\sqrt289}{2.4} \\
& = \frac{-32}{8} & & = \frac{2}{8} \\
& = -4 & & = \frac{1}{4} \\ \\ V &= \Big\{ -4 ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \((2x+1)(5x+2)-x(9x+2)=-14\\
\Leftrightarrow 10x^2+4x+5x+2 -9x^2-2x+14=0 \\
\Leftrightarrow x^2+4x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+4x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (4)^2-4.1.16 & &\\
& = 16-64 & & \\
& = -48 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(\frac{1}{2}x^2+\frac{25}{24}x+\frac{1}{2}=0\\
\Leftrightarrow \color{red}{24.} \left(\frac{1}{2}x^2+\frac{25}{24}x+\frac{1}{2}\right)=0 \color{red}{.24} \\
\Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.12.12 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\
& = \frac{-32}{24} & & = \frac{-18}{24} \\
& = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x(x+13)=15(x+1) \\
\Leftrightarrow x^2+13x=15x+15 \\
\Leftrightarrow x^2-2x-15=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x-15=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-2)^2-4.1.(-15) & &\\
& = 4+60 & & \\
& = 64 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-2)-\sqrt64}{2.1} & & = \frac{-(-2)+\sqrt64}{2.1} \\
& = \frac{-6}{2} & & = \frac{10}{2} \\
& = -3 & & = 5 \\ \\ V &= \Big\{ -3 ; 5 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2+\frac{13}{6}x+\frac{1}{2}=0\\
\Leftrightarrow \color{red}{6.} \left(2x^2+\frac{13}{6}x+\frac{1}{2}\right)=0 \color{red}{.6} \\
\Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.12.3 & &\\
& = 169-144 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\
& = \frac{-18}{24} & & = \frac{-8}{24} \\
& = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2+\frac{7}{12}x-3=0\\
\Leftrightarrow \color{red}{12.} \left(\frac{1}{3}x^2+\frac{7}{12}x-3\right)=0 \color{red}{.12} \\
\Leftrightarrow 4x^2+7x-36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+7x-36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.4.(-36) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.4} & & = \frac{-7+\sqrt625}{2.4} \\
& = \frac{-32}{8} & & = \frac{18}{8} \\
& = -4 & & = \frac{9}{4} \\ \\ V &= \Big\{ -4 ; \frac{9}{4} \Big\} & &\end{align} \\ -----------------\)
- \(-(5-21x)=-x^2-(-1-20x) \\
\Leftrightarrow -5+21x=-x^2+1+20x \\
\Leftrightarrow x^2+x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+x-6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (1)^2-4.1.(-6) & &\\
& = 1+24 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-1-\sqrt25}{2.1} & & = \frac{-1+\sqrt25}{2.1} \\
& = \frac{-6}{2} & & = \frac{4}{2} \\
& = -3 & & = 2 \\ \\ V &= \Big\{ -3 ; 2 \Big\} & &\end{align} \\ -----------------\)