Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(5x^2-(11x-1)=x(x-16)\)
- \((-x-5)(x-1)-x(-13x-19)=-7\)
- \(x(x-8)=4(x-9)\)
- \(\frac{1}{5}x^2+\frac{13}{5}x+\frac{42}{5}=0\)
- \(-(7-29x)=-4x^2-(43-5x)\)
- \(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}=0\)
- \((x-4)(-x-4)-x(-9x-13)=-2\)
- \(-\frac{1}{3}x=-\frac{1}{15}x^2+\frac{10}{3}\)
- \(3x^2-(18x+18)=x(x-23)\)
- \(5x^2-(18x-36)=x(x+6)\)
- \(-(5-37x)=-72x^2-(7-12x)\)
- \(11x^2-(14x-144)=2x(x-43)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(5x^2-(11x-1)=x(x-16) \\
\Leftrightarrow 5x^2-11x+1=x^2-16x \\
\Leftrightarrow 4x^2+5x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.4.1 & &\\
& = 25-16 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt9}{2.4} & & = \frac{-5+\sqrt9}{2.4} \\
& = \frac{-8}{8} & & = \frac{-2}{8} \\
& = -1 & & = \frac{-1}{4} \\ \\ V &= \Big\{ -1 ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
- \((-x-5)(x-1)-x(-13x-19)=-7\\
\Leftrightarrow -x^2+x-5x+5 +13x^2+19x+7=0 \\
\Leftrightarrow 12x^2+25x+12=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+25x+12=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.12.12 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.12} & & = \frac{-25+\sqrt49}{2.12} \\
& = \frac{-32}{24} & & = \frac{-18}{24} \\
& = \frac{-4}{3} & & = \frac{-3}{4} \\ \\ V &= \Big\{ \frac{-4}{3} ; \frac{-3}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x(x-8)=4(x-9) \\
\Leftrightarrow x^2-8x=4x-36 \\
\Leftrightarrow x^2-12x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-12x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-12)^2-4.1.36 & &\\
& = 144-144 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-12)}{2.1} & & \\
& = 6 & & \\V &= \Big\{ 6 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{5}x^2+\frac{13}{5}x+\frac{42}{5}=0\\
\Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2+\frac{13}{5}x+\frac{42}{5}\right)=0 \color{red}{.5} \\
\Leftrightarrow x^2+13x+42=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+13x+42=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (13)^2-4.1.42 & &\\
& = 169-168 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-13-\sqrt1}{2.1} & & = \frac{-13+\sqrt1}{2.1} \\
& = \frac{-14}{2} & & = \frac{-12}{2} \\
& = -7 & & = -6 \\ \\ V &= \Big\{ -7 ; -6 \Big\} & &\end{align} \\ -----------------\)
- \(-(7-29x)=-4x^2-(43-5x) \\
\Leftrightarrow -7+29x=-4x^2-43+5x \\
\Leftrightarrow 4x^2+24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+24x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (24)^2-4.4.36 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-24}{2.4} & & \\
& = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}=0\\
\Leftrightarrow \color{red}{16.} \left(\frac{9}{4}x^2+\frac{7}{16}x-\frac{1}{4}\right)=0 \color{red}{.16} \\
\Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.36.(-4) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\
& = \frac{-32}{72} & & = \frac{18}{72} \\
& = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
- \((x-4)(-x-4)-x(-9x-13)=-2\\
\Leftrightarrow -x^2-4x+4x+16 +9x^2+13x+2=0 \\
\Leftrightarrow 8x^2+25x+18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+25x+18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.8.18 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.8} & & = \frac{-25+\sqrt49}{2.8} \\
& = \frac{-32}{16} & & = \frac{-18}{16} \\
& = -2 & & = \frac{-9}{8} \\ \\ V &= \Big\{ -2 ; \frac{-9}{8} \Big\} & &\end{align} \\ -----------------\)
- \(-\frac{1}{3}x=-\frac{1}{15}x^2+\frac{10}{3} \\
\Leftrightarrow \frac{1}{15}x^2-\frac{1}{3}x-\frac{10}{3}=0 \\
\Leftrightarrow \color{red}{15.} \left(\frac{1}{15}x^2-\frac{1}{3}x-\frac{10}{3}\right)=0 \color{red}{.15} \\
\Leftrightarrow x^2-5x-50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-5x-50=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-5)^2-4.1.(-50) & &\\
& = 25+200 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-5)-\sqrt225}{2.1} & & = \frac{-(-5)+\sqrt225}{2.1} \\
& = \frac{-10}{2} & & = \frac{20}{2} \\
& = -5 & & = 10 \\ \\ V &= \Big\{ -5 ; 10 \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(18x+18)=x(x-23) \\
\Leftrightarrow 3x^2-18x-18=x^2-23x \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(5x^2-(18x-36)=x(x+6) \\
\Leftrightarrow 5x^2-18x+36=x^2+6x \\
\Leftrightarrow 4x^2-24x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-24x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.4.36 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.4} & & \\
& = 3 & & \\V &= \Big\{ 3 \Big\} & &\end{align} \\ -----------------\)
- \(-(5-37x)=-72x^2-(7-12x) \\
\Leftrightarrow -5+37x=-72x^2-7+12x \\
\Leftrightarrow 72x^2+25x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{72x^2+25x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.72.2 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.72} & & = \frac{-25+\sqrt49}{2.72} \\
& = \frac{-32}{144} & & = \frac{-18}{144} \\
& = \frac{-2}{9} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-2}{9} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(11x^2-(14x-144)=2x(x-43) \\
\Leftrightarrow 11x^2-14x+144=2x^2-86x \\
\Leftrightarrow 9x^2+72x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+72x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (72)^2-4.9.144 & &\\
& = 5184-5184 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-72}{2.9} & & \\
& = -4 & & \\V &= \Big\{ -4 \Big\} & &\end{align} \\ -----------------\)