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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(x(4x-23)=-25(x+1)\)
2. \(x(6x+11)=6(x+1)\)
3. \(\frac{1}{2}x^2-\frac{1}{2}x+\frac{9}{32}=0\)
4. \((-4x+2)(-4x-1)-x(15x+12)=-146\)
5. \(x(2x+79)=72(x+1)\)
6. \((x-5)(-4x+2)-x(-5x-6)=-46\)
7. \(38x^2-(17x+4)=2x(x-12)\)
8. \(-(12-x)=-x^2-(28-9x)\)
9. \((-4x+4)(3x-2)-x(-28x+48)=-44\)
10. \(x(9x+1)=-(x+1)\)
11. \(x(16x-60)=4(x-16)\)
12. \(18x^2-(19x+3)=6x(x-4)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(x(4x-23)=-25(x+1) \\ \Leftrightarrow 4x^2-23x=-25x-25 \\ \Leftrightarrow 4x^2+2x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+2x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.4.25 & &\\ & = 4-400 & & \\ & = -396 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
2. \(x(6x+11)=6(x+1) \\ \Leftrightarrow 6x^2+11x=6x+6 \\ \Leftrightarrow 6x^2+5x-6=0 \\\text{We zoeken de oplossingen van } \color{blue}{6x^2+5x-6=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.6.(-6) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.6} & & = \frac{-5+\sqrt169}{2.6} \\ & = \frac{-18}{12} & & = \frac{8}{12} \\ & = \frac{-3}{2} & & = \frac{2}{3} \\ \\ V &= \Big\{ \frac{-3}{2} ; \frac{2}{3} \Big\} & &\end{align} \\ -----------------\)
3. \(\frac{1}{2}x^2-\frac{1}{2}x+\frac{9}{32}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{2}x^2-\frac{1}{2}x+\frac{9}{32}\right)=0 \color{red}{.32} \\ \Leftrightarrow 16x^2-16x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-16x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.16.9 & &\\ & = 256-576 & & \\ & = -320 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \((-4x+2)(-4x-1)-x(15x+12)=-146\\ \Leftrightarrow 16x^2+4x-8x-2 -15x^2-12x+146=0 \\ \Leftrightarrow x^2-10x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+144=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-10)^2-4.1.144 & &\\ & = 100-576 & & \\ & = -476 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(x(2x+79)=72(x+1) \\ \Leftrightarrow 2x^2+79x=72x+72 \\ \Leftrightarrow 2x^2+7x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+7x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.2.(-72) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.2} & & = \frac{-7+\sqrt625}{2.2} \\ & = \frac{-32}{4} & & = \frac{18}{4} \\ & = -8 & & = \frac{9}{2} \\ \\ V &= \Big\{ -8 ; \frac{9}{2} \Big\} & &\end{align} \\ -----------------\)
6. \((x-5)(-4x+2)-x(-5x-6)=-46\\ \Leftrightarrow -4x^2+2x+20x-10 +5x^2+6x+46=0 \\ \Leftrightarrow x^2-2x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-2x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.1.36 & &\\ & = 4-144 & & \\ & = -140 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(38x^2-(17x+4)=2x(x-12) \\ \Leftrightarrow 38x^2-17x-4=2x^2-24x \\ \Leftrightarrow 36x^2+7x-4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+7x-4=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (7)^2-4.36.(-4) & &\\ & = 49+576 & & \\ & = 625 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-7-\sqrt625}{2.36} & & = \frac{-7+\sqrt625}{2.36} \\ & = \frac{-32}{72} & & = \frac{18}{72} \\ & = \frac{-4}{9} & & = \frac{1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
8. \(-(12-x)=-x^2-(28-9x) \\ \Leftrightarrow -12+x=-x^2-28+9x \\ \Leftrightarrow x^2-8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-8)^2-4.1.16 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-8)}{2.1} & & \\ & = 4 & & \\V &= \Big\{ 4 \Big\} & &\end{align} \\ -----------------\)
9. \((-4x+4)(3x-2)-x(-28x+48)=-44\\ \Leftrightarrow -12x^2+8x+12x-8 +28x^2-48x+44=0 \\ \Leftrightarrow 16x^2-48x+36=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-48x+36=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-48)^2-4.16.36 & &\\ & = 2304-2304 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-48)}{2.16} & & \\ & = \frac{3}{2} & & \\V &= \Big\{ \frac{3}{2} \Big\} & &\end{align} \\ -----------------\)
10. \(x(9x+1)=-(x+1) \\ \Leftrightarrow 9x^2+x=-x-1 \\ \Leftrightarrow 9x^2+2x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+2x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (2)^2-4.9.1 & &\\ & = 4-36 & & \\ & = -32 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
11. \(x(16x-60)=4(x-16) \\ \Leftrightarrow 16x^2-60x=4x-64 \\ \Leftrightarrow 16x^2-64x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-64x+64=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-64)^2-4.16.64 & &\\ & = 4096-4096 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-64)}{2.16} & & \\ & = 2 & & \\V &= \Big\{ 2 \Big\} & &\end{align} \\ -----------------\)
12. \(18x^2-(19x+3)=6x(x-4) \\ \Leftrightarrow 18x^2-19x-3=6x^2-24x \\ \Leftrightarrow 12x^2+5x-3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+5x-3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.12.(-3) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.12} & & = \frac{-5+\sqrt169}{2.12} \\ & = \frac{-18}{24} & & = \frac{8}{24} \\ & = \frac{-3}{4} & & = \frac{1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)