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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(-(3-17x)=-4x^2-(4-13x)\)
2. \((2x-1)(2x+5)-x(0x+31)=-54\)
3. \(-(11-33x)=-8x^2-(13-16x)\)
4. \(-(11+8x)=-x^2-(92-10x)\)
5. \((-3x+1)(4x-3)-x(-28x+24)=-12\)
6. \(-(15-24x)=-16x^2-(16-14x)\)
7. \(\frac{1}{4}x^2-x+\frac{25}{4}=0\)
8. \(13x^2-(9x-3)=x(x-22)\)
9. \(2x^2-(8x+72)=x(x-7)\)
10. \(x(9x-7)=-(x+1)\)
11. \(x(2x+23)=8(x+1)\)
12. \(-(3+11x)=-x^2-(53-4x)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(-(3-17x)=-4x^2-(4-13x) \\ \Leftrightarrow -3+17x=-4x^2-4+13x \\ \Leftrightarrow 4x^2+4x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (4)^2-4.4.1 & &\\ & = 16-16 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-4}{2.4} & & \\ & = -\frac{1}{2} & & \\V &= \Big\{ -\frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
2. \((2x-1)(2x+5)-x(0x+31)=-54\\ \Leftrightarrow 4x^2+10x-2x-5 +0x^2-31x+54=0 \\ \Leftrightarrow 4x^2-26x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-26x+49=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-26)^2-4.4.49 & &\\ & = 676-784 & & \\ & = -108 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
3. \(-(11-33x)=-8x^2-(13-16x) \\ \Leftrightarrow -11+33x=-8x^2-13+16x \\ \Leftrightarrow 8x^2+17x+2=0 \\\text{We zoeken de oplossingen van } \color{blue}{8x^2+17x+2=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (17)^2-4.8.2 & &\\ & = 289-64 & & \\ & = 225 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-17-\sqrt225}{2.8} & & = \frac{-17+\sqrt225}{2.8} \\ & = \frac{-32}{16} & & = \frac{-2}{16} \\ & = -2 & & = \frac{-1}{8} \\ \\ V &= \Big\{ -2 ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
4. \(-(11+8x)=-x^2-(92-10x) \\ \Leftrightarrow -11-8x=-x^2-92+10x \\ \Leftrightarrow x^2-18x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-18x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.1.81 & &\\ & = 324-324 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-18)}{2.1} & & \\ & = 9 & & \\V &= \Big\{ 9 \Big\} & &\end{align} \\ -----------------\)
5. \((-3x+1)(4x-3)-x(-28x+24)=-12\\ \Leftrightarrow -12x^2+9x+4x-3 +28x^2-24x+12=0 \\ \Leftrightarrow 16x^2-18x+9=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2-18x+9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-18)^2-4.16.9 & &\\ & = 324-576 & & \\ & = -252 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
6. \(-(15-24x)=-16x^2-(16-14x) \\ \Leftrightarrow -15+24x=-16x^2-16+14x \\ \Leftrightarrow 16x^2+10x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+10x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (10)^2-4.16.1 & &\\ & = 100-64 & & \\ & = 36 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-10-\sqrt36}{2.16} & & = \frac{-10+\sqrt36}{2.16} \\ & = \frac{-16}{32} & & = \frac{-4}{32} \\ & = \frac{-1}{2} & & = \frac{-1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{-1}{8} \Big\} & &\end{align} \\ -----------------\)
7. \(\frac{1}{4}x^2-x+\frac{25}{4}=0\\ \Leftrightarrow \color{red}{4.} \left(\frac{1}{4}x^2-x+\frac{25}{4}\right)=0 \color{red}{.4} \\ \Leftrightarrow 4x^2-16x+100=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-16x+100=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-16)^2-4.4.100 & &\\ & = 256-1600 & & \\ & = -1344 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(13x^2-(9x-3)=x(x-22) \\ \Leftrightarrow 13x^2-9x+3=x^2-22x \\ \Leftrightarrow 12x^2+13x+3=0 \\\text{We zoeken de oplossingen van } \color{blue}{12x^2+13x+3=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (13)^2-4.12.3 & &\\ & = 169-144 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-13-\sqrt25}{2.12} & & = \frac{-13+\sqrt25}{2.12} \\ & = \frac{-18}{24} & & = \frac{-8}{24} \\ & = \frac{-3}{4} & & = \frac{-1}{3} \\ \\ V &= \Big\{ \frac{-3}{4} ; \frac{-1}{3} \Big\} & &\end{align} \\ -----------------\)
9. \(2x^2-(8x+72)=x(x-7) \\ \Leftrightarrow 2x^2-8x-72=x^2-7x \\ \Leftrightarrow x^2-x-72=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-72=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-72) & &\\ & = 1+288 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt289}{2.1} & & = \frac{-(-1)+\sqrt289}{2.1} \\ & = \frac{-16}{2} & & = \frac{18}{2} \\ & = -8 & & = 9 \\ \\ V &= \Big\{ -8 ; 9 \Big\} & &\end{align} \\ -----------------\)
10. \(x(9x-7)=-(x+1) \\ \Leftrightarrow 9x^2-7x=-x-1 \\ \Leftrightarrow 9x^2-6x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2-6x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.9.1 & &\\ & = 36-36 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-(-6)}{2.9} & & \\ & = \frac{1}{3} & & \\V &= \Big\{ \frac{1}{3} \Big\} & &\end{align} \\ -----------------\)
11. \(x(2x+23)=8(x+1) \\ \Leftrightarrow 2x^2+23x=8x+8 \\ \Leftrightarrow 2x^2+15x-8=0 \\\text{We zoeken de oplossingen van } \color{blue}{2x^2+15x-8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (15)^2-4.2.(-8) & &\\ & = 225+64 & & \\ & = 289 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-15-\sqrt289}{2.2} & & = \frac{-15+\sqrt289}{2.2} \\ & = \frac{-32}{4} & & = \frac{2}{4} \\ & = -8 & & = \frac{1}{2} \\ \\ V &= \Big\{ -8 ; \frac{1}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(-(3+11x)=-x^2-(53-4x) \\ \Leftrightarrow -3-11x=-x^2-53+4x \\ \Leftrightarrow x^2-15x+50=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-15x+50=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-15)^2-4.1.50 & &\\ & = 225-200 & & \\ & = 25 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-15)-\sqrt25}{2.1} & & = \frac{-(-15)+\sqrt25}{2.1} \\ & = \frac{10}{2} & & = \frac{20}{2} \\ & = 5 & & = 10 \\ \\ V &= \Big\{ 5 ; 10 \Big\} & &\end{align} \\ -----------------\)