Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(x(x+109)=110(x+1)\)
- \(2x^2-(8x-88)=x(x+11)\)
- \((-3x-5)(3x+5)-x(-25x-46)=-24\)
- \((-2x-2)(-x-3)-x(-6x+5)=24\)
- \(-(13+12x)=-x^2-(97-7x)\)
- \((-x-5)(-3x-2)-x(-6x-36)=-54\)
- \((-3x-2)(-4x+5)-x(11x-24)=122\)
- \((2x-4)(-4x-3)-x(-24x+70)=-52\)
- \((-5x+2)(-3x+1)-x(14x-5)=122\)
- \((5x+4)(2x-2)-x(x+6)=-24\)
- \((-5x-2)(-3x+1)-x(14x-1)=-38\)
- \((-2x+2)(4x-3)-x(-9x+24)=-150\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(x(x+109)=110(x+1) \\
\Leftrightarrow x^2+109x=110x+110 \\
\Leftrightarrow x^2-x-110=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-110=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-1)^2-4.1.(-110) & &\\
& = 1+440 & & \\
& = 441 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-1)-\sqrt441}{2.1} & & = \frac{-(-1)+\sqrt441}{2.1} \\
& = \frac{-20}{2} & & = \frac{22}{2} \\
& = -10 & & = 11 \\ \\ V &= \Big\{ -10 ; 11 \Big\} & &\end{align} \\ -----------------\)
- \(2x^2-(8x-88)=x(x+11) \\
\Leftrightarrow 2x^2-8x+88=x^2+11x \\
\Leftrightarrow x^2-19x+88=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+88=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-19)^2-4.1.88 & &\\
& = 361-352 & & \\
& = 9 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-19)-\sqrt9}{2.1} & & = \frac{-(-19)+\sqrt9}{2.1} \\
& = \frac{16}{2} & & = \frac{22}{2} \\
& = 8 & & = 11 \\ \\ V &= \Big\{ 8 ; 11 \Big\} & &\end{align} \\ -----------------\)
- \((-3x-5)(3x+5)-x(-25x-46)=-24\\
\Leftrightarrow -9x^2-15x-15x-25 +25x^2+46x+24=0 \\
\Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (6)^2-4.16.(-1) & &\\
& = 36+64 & & \\
& = 100 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\
& = \frac{-16}{32} & & = \frac{4}{32} \\
& = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
- \((-2x-2)(-x-3)-x(-6x+5)=24\\
\Leftrightarrow 2x^2+6x+2x+6 +6x^2-5x-24=0 \\
\Leftrightarrow 8x^2+7x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+7x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (7)^2-4.8.(-18) & &\\
& = 49+576 & & \\
& = 625 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-7-\sqrt625}{2.8} & & = \frac{-7+\sqrt625}{2.8} \\
& = \frac{-32}{16} & & = \frac{18}{16} \\
& = -2 & & = \frac{9}{8} \\ \\ V &= \Big\{ -2 ; \frac{9}{8} \Big\} & &\end{align} \\ -----------------\)
- \(-(13+12x)=-x^2-(97-7x) \\
\Leftrightarrow -13-12x=-x^2-97+7x \\
\Leftrightarrow x^2-19x+84=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-19x+84=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-19)^2-4.1.84 & &\\
& = 361-336 & & \\
& = 25 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-19)-\sqrt25}{2.1} & & = \frac{-(-19)+\sqrt25}{2.1} \\
& = \frac{14}{2} & & = \frac{24}{2} \\
& = 7 & & = 12 \\ \\ V &= \Big\{ 7 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \((-x-5)(-3x-2)-x(-6x-36)=-54\\
\Leftrightarrow 3x^2+2x+15x+10 +6x^2+36x+54=0 \\
\Leftrightarrow 9x^2+48x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+48x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (48)^2-4.9.64 & &\\
& = 2304-2304 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-48}{2.9} & & \\
& = -\frac{8}{3} & & \\V &= \Big\{ -\frac{8}{3} \Big\} & &\end{align} \\ -----------------\)
- \((-3x-2)(-4x+5)-x(11x-24)=122\\
\Leftrightarrow 12x^2-15x+8x-10 -11x^2+24x-122=0 \\
\Leftrightarrow x^2-x-132=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-132=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-1)^2-4.1.(-132) & &\\
& = 1+528 & & \\
& = 529 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-1)-\sqrt529}{2.1} & & = \frac{-(-1)+\sqrt529}{2.1} \\
& = \frac{-22}{2} & & = \frac{24}{2} \\
& = -11 & & = 12 \\ \\ V &= \Big\{ -11 ; 12 \Big\} & &\end{align} \\ -----------------\)
- \((2x-4)(-4x-3)-x(-24x+70)=-52\\
\Leftrightarrow -8x^2-6x+16x+12 +24x^2-70x+52=0 \\
\Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-64)^2-4.16.64 & &\\
& = 4096-4096 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-64)}{2.16} & & \\
& = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
- \((-5x+2)(-3x+1)-x(14x-5)=122\\
\Leftrightarrow 15x^2-5x-6x+2 -14x^2+5x-122=0 \\
\Leftrightarrow x^2+2x-120=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+2x-120=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (2)^2-4.1.(-120) & &\\
& = 4+480 & & \\
& = 484 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-2-\sqrt484}{2.1} & & = \frac{-2+\sqrt484}{2.1} \\
& = \frac{-24}{2} & & = \frac{20}{2} \\
& = -12 & & = 10 \\ \\ V &= \Big\{ -12 ; 10 \Big\} & &\end{align} \\ -----------------\)
- \((5x+4)(2x-2)-x(x+6)=-24\\
\Leftrightarrow 10x^2-10x+8x-8 -x^2-6x+24=0 \\
\Leftrightarrow 9x^2-24x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-24x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.9.16 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.9} & & \\
& = \frac{4}{3} & & \\V &= \Big\{ \frac{4}{3} \Big\} & &\end{align} \\ -----------------\)
- \((-5x-2)(-3x+1)-x(14x-1)=-38\\
\Leftrightarrow 15x^2-5x+6x-2 -14x^2+x+38=0 \\
\Leftrightarrow x^2-6x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+36=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-6)^2-4.1.36 & &\\
& = 36-144 & & \\
& = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \((-2x+2)(4x-3)-x(-9x+24)=-150\\
\Leftrightarrow -8x^2+6x+8x-6 +9x^2-24x+150=0 \\
\Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.1.144 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.1} & & \\
& = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)