Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(17x^2-(8x-144)=x(x-104)\)
- \(\frac{1}{5}x^2+x+\frac{5}{4}=0\)
- \(3x^2-(16x+8)=x(x-31)\)
- \(-(13-24x)=-8x^2-(11-9x)\)
- \(12x^2-(10x-144)=3x(x-2)\)
- \(-(10-15x)=-16x^2-(11-5x)\)
- \(x(24x+27)=2(x-3)\)
- \(\frac{9}{35}x^2+\frac{6}{5}x+\frac{7}{5}=0\)
- \((-3x+3)(-4x+5)-x(10x-5)=33\)
- \(\frac{17}{3}x=-\frac{16}{3}x^2-\frac{1}{3}\)
- \(2x^2-(5x-2)=x(x-2)\)
- \(x(x-79)=-63(x+1)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(17x^2-(8x-144)=x(x-104) \\
\Leftrightarrow 17x^2-8x+144=x^2-104x \\
\Leftrightarrow 16x^2+96x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+96x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (96)^2-4.16.144 & &\\
& = 9216-9216 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-96}{2.16} & & \\
& = -3 & & \\V &= \Big\{ -3 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{5}x^2+x+\frac{5}{4}=0\\
\Leftrightarrow \color{red}{20.} \left(\frac{1}{5}x^2+x+\frac{5}{4}\right)=0 \color{red}{.20} \\
\Leftrightarrow 4x^2+20x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+20x+25=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (20)^2-4.4.25 & &\\
& = 400-400 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-20}{2.4} & & \\
& = -\frac{5}{2} & & \\V &= \Big\{ -\frac{5}{2} \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(16x+8)=x(x-31) \\
\Leftrightarrow 3x^2-16x-8=x^2-31x \\
\Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.2.(-8) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\
& = \frac{-32}{4} & & = \frac{2}{4} \\
& = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
- \(-(13-24x)=-8x^2-(11-9x) \\
\Leftrightarrow -13+24x=-8x^2-11+9x \\
\Leftrightarrow 8x^2+15x-2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+15x-2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.8.(-2) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.8} & & = \frac{-15+\sqrt289}{2.8} \\
& = \frac{-32}{16} & & = \frac{2}{16} \\
& = -2 & & = \frac{1}{8} \\ \\ V &= \Big\{ -2 ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(12x^2-(10x-144)=3x(x-2) \\
\Leftrightarrow 12x^2-10x+144=3x^2-6x \\
\Leftrightarrow 9x^2-4x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-4x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-4)^2-4.9.144 & &\\
& = 16-5184 & & \\
& = -5168 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(-(10-15x)=-16x^2-(11-5x) \\
\Leftrightarrow -10+15x=-16x^2-11+5x \\
\Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (10)^2-4.16.1 & &\\
& = 100-64 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\
& = \frac{-16}{32} & & = \frac{-4}{32} \\
& = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
- \(x(24x+27)=2(x-3) \\
\Leftrightarrow 24x^2+27x=2x-6 \\
\Leftrightarrow 24x^2+25x+6=0 \\\text{We zoeken de oplossingen van } \color{blue}{24x^2+25x+6=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.24.6 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.24} & & = \frac{-25+\sqrt49}{2.24} \\
& = \frac{-32}{48} & & = \frac{-18}{48} \\
& = \frac{-2}{3} & & = \frac{-3}{8} \\ \\ V &= \Big\{ \frac{-2}{3} ; \frac{-3}{8} \Big\} & &\end{align} \\ -----------------\)
- \(\frac{9}{35}x^2+\frac{6}{5}x+\frac{7}{5}=0\\
\Leftrightarrow \color{red}{35.} \left(\frac{9}{35}x^2+\frac{6}{5}x+\frac{7}{5}\right)=0 \color{red}{.35} \\
\Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (42)^2-4.9.49 & &\\
& = 1764-1764 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-42}{2.9} & & \\
& = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
- \((-3x+3)(-4x+5)-x(10x-5)=33\\
\Leftrightarrow 12x^2-15x-12x+15 -10x^2+5x-33=0 \\
\Leftrightarrow 2x^2+5x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+5x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (5)^2-4.2.(-18) & &\\
& = 25+144 & & \\
& = 169 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-5-\sqrt169}{2.2} & & = \frac{-5+\sqrt169}{2.2} \\
& = \frac{-18}{4} & & = \frac{8}{4} \\
& = \frac{-9}{2} & & = 2 \\ \\ V &= \Big\{ \frac{-9}{2} ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{17}{3}x=-\frac{16}{3}x^2-\frac{1}{3} \\
\Leftrightarrow \frac{16}{3}x^2+\frac{17}{3}x+\frac{1}{3}=0 \\
\Leftrightarrow \color{red}{3.} \left(\frac{16}{3}x^2+\frac{17}{3}x+\frac{1}{3}\right)=0 \color{red}{.3} \\
\Leftrightarrow 16x^2+17x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+17x+1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (17)^2-4.16.1 & &\\
& = 289-64 & & \\
& = 225 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-17-\sqrt225}{2.16} & & = \frac{-17+\sqrt225}{2.16} \\
& = \frac{-32}{32} & & = \frac{-2}{32} \\
& = -1 & & = \frac{-1}{16} \\ \\ V &= \Big\{ -1 ; \frac{-1}{16} \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(5x-2)=x(x-2) \\
\Leftrightarrow 2x^2-5x+2=x^2-2x \\
\Leftrightarrow x^2-3x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-3x+2=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-3)^2-4.1.2 & &\\
& = 9-8 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-3)-\sqrt1}{2.1} & & = \frac{-(-3)+\sqrt1}{2.1} \\
& = \frac{2}{2} & & = \frac{4}{2} \\
& = 1 & & = 2 \\ \\ V &= \Big\{ 1 ; 2 \Big\} & &\end{align} \\ -----------------\)
- \(x(x-79)=-63(x+1) \\
\Leftrightarrow x^2-79x=-63x-63 \\
\Leftrightarrow x^2-16x+63=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-16x+63=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-16)^2-4.1.63 & &\\
& = 256-252 & & \\
& = 4 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-16)-\sqrt4}{2.1} & & = \frac{-(-16)+\sqrt4}{2.1} \\
& = \frac{14}{2} & & = \frac{18}{2} \\
& = 7 & & = 9 \\ \\ V &= \Big\{ 7 ; 9 \Big\} & &\end{align} \\ -----------------\)