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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

1. \(\frac{1}{2}x^2+\frac{1}{4}x+\frac{1}{32}=0\)
2. \((-3x+1)(2x+5)-x(-7x+3)=-17\)
3. \((4x-5)(-5x-4)-x(-21x+10)=-5\)
4. \(\frac{9}{2}x^2+11x+\frac{25}{2}=0\)
5. \(-\frac{1}{5}x=-\frac{1}{5}x^2+\frac{12}{5}\)
6. \((3x+4)(4x-2)-x(8x-2)=-89\)
7. \(-(2-17x)=-4x^2-(27-19x)\)
8. \(-(7-26x)=-4x^2-(23-18x)\)
9. \((-5x+1)(x-2)-x(-21x+2)=-1\)
10. \(-(7-23x)=-4x^2-(-2-18x)\)
11. \(x(18x+17)=-8(x+1)\)
12. \(2x^2-(7x-30)=x(x+4)\)

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Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen

Verbetersleutel

1. \(\frac{1}{2}x^2+\frac{1}{4}x+\frac{1}{32}=0\\ \Leftrightarrow \color{red}{32.} \left(\frac{1}{2}x^2+\frac{1}{4}x+\frac{1}{32}\right)=0 \color{red}{.32} \\ \Leftrightarrow 16x^2+8x+1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+8x+1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.16.1 & &\\ & = 64-64 & & \\ & = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\ & = \frac{-8}{2.16} & & \\ & = -\frac{1}{4} & & \\V &= \Big\{ -\frac{1}{4} \Big\} & &\end{align} \\ -----------------\)
2. \((-3x+1)(2x+5)-x(-7x+3)=-17\\ \Leftrightarrow -6x^2-15x+2x+5 +7x^2-3x+17=0 \\ \Leftrightarrow x^2-13x+22=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-13x+22=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-13)^2-4.1.22 & &\\ & = 169-88 & & \\ & = 81 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-13)-\sqrt81}{2.1} & & = \frac{-(-13)+\sqrt81}{2.1} \\ & = \frac{4}{2} & & = \frac{22}{2} \\ & = 2 & & = 11 \\ \\ V &= \Big\{ 2 ; 11 \Big\} & &\end{align} \\ -----------------\)
3. \((4x-5)(-5x-4)-x(-21x+10)=-5\\ \Leftrightarrow -20x^2-16x+25x+20 +21x^2-10x+5=0 \\ \Leftrightarrow x^2-6x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-6x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-6)^2-4.1.25 & &\\ & = 36-100 & & \\ & = -64 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
4. \(\frac{9}{2}x^2+11x+\frac{25}{2}=0\\ \Leftrightarrow \color{red}{2.} \left(\frac{9}{2}x^2+11x+\frac{25}{2}\right)=0 \color{red}{.2} \\ \Leftrightarrow 9x^2+22x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+22x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (22)^2-4.9.25 & &\\ & = 484-900 & & \\ & = -416 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
5. \(-\frac{1}{5}x=-\frac{1}{5}x^2+\frac{12}{5} \\ \Leftrightarrow \frac{1}{5}x^2-\frac{1}{5}x-\frac{12}{5}=0 \\ \Leftrightarrow \color{red}{5.} \left(\frac{1}{5}x^2-\frac{1}{5}x-\frac{12}{5}\right)=0 \color{red}{.5} \\ \Leftrightarrow x^2-x-12=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-12=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-1)^2-4.1.(-12) & &\\ & = 1+48 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-1)-\sqrt49}{2.1} & & = \frac{-(-1)+\sqrt49}{2.1} \\ & = \frac{-6}{2} & & = \frac{8}{2} \\ & = -3 & & = 4 \\ \\ V &= \Big\{ -3 ; 4 \Big\} & &\end{align} \\ -----------------\)
6. \((3x+4)(4x-2)-x(8x-2)=-89\\ \Leftrightarrow 12x^2-6x+16x-8 -8x^2+2x+89=0 \\ \Leftrightarrow 4x^2-12x+81=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-12x+81=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-12)^2-4.4.81 & &\\ & = 144-1296 & & \\ & = -1152 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
7. \(-(2-17x)=-4x^2-(27-19x) \\ \Leftrightarrow -2+17x=-4x^2-27+19x \\ \Leftrightarrow 4x^2-2x+25=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2-2x+25=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-2)^2-4.4.25 & &\\ & = 4-400 & & \\ & = -396 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
8. \(-(7-26x)=-4x^2-(23-18x) \\ \Leftrightarrow -7+26x=-4x^2-23+18x \\ \Leftrightarrow 4x^2+8x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+8x+16=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (8)^2-4.4.16 & &\\ & = 64-256 & & \\ & = -192 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
9. \((-5x+1)(x-2)-x(-21x+2)=-1\\ \Leftrightarrow -5x^2+10x+x-2 +21x^2-2x+1=0 \\ \Leftrightarrow 16x^2+6x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+6x-1=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (6)^2-4.16.(-1) & &\\ & = 36+64 & & \\ & = 100 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-6-\sqrt100}{2.16} & & = \frac{-6+\sqrt100}{2.16} \\ & = \frac{-16}{32} & & = \frac{4}{32} \\ & = \frac{-1}{2} & & = \frac{1}{8} \\ \\ V &= \Big\{ \frac{-1}{2} ; \frac{1}{8} \Big\} & &\end{align} \\ -----------------\)
10. \(-(7-23x)=-4x^2-(-2-18x) \\ \Leftrightarrow -7+23x=-4x^2+2+18x \\ \Leftrightarrow 4x^2+5x-9=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+5x-9=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (5)^2-4.4.(-9) & &\\ & = 25+144 & & \\ & = 169 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-5-\sqrt169}{2.4} & & = \frac{-5+\sqrt169}{2.4} \\ & = \frac{-18}{8} & & = \frac{8}{8} \\ & = \frac{-9}{4} & & = 1 \\ \\ V &= \Big\{ \frac{-9}{4} ; 1 \Big\} & &\end{align} \\ -----------------\)
11. \(x(18x+17)=-8(x+1) \\ \Leftrightarrow 18x^2+17x=-8x-8 \\ \Leftrightarrow 18x^2+25x+8=0 \\\text{We zoeken de oplossingen van } \color{blue}{18x^2+25x+8=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (25)^2-4.18.8 & &\\ & = 625-576 & & \\ & = 49 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-25-\sqrt49}{2.18} & & = \frac{-25+\sqrt49}{2.18} \\ & = \frac{-32}{36} & & = \frac{-18}{36} \\ & = \frac{-8}{9} & & = \frac{-1}{2} \\ \\ V &= \Big\{ \frac{-8}{9} ; \frac{-1}{2} \Big\} & &\end{align} \\ -----------------\)
12. \(2x^2-(7x-30)=x(x+4) \\ \Leftrightarrow 2x^2-7x+30=x^2+4x \\ \Leftrightarrow x^2-11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+30=0} \\ \\\begin{align} D & = b^2 - 4.a.c & & \\ & = (-11)^2-4.1.30 & &\\ & = 121-120 & & \\ & = 1 & & \\ \\ x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\ & = \frac{-(-11)-\sqrt1}{2.1} & & = \frac{-(-11)+\sqrt1}{2.1} \\ & = \frac{10}{2} & & = \frac{12}{2} \\ & = 5 & & = 6 \\ \\ V &= \Big\{ 5 ; 6 \Big\} & &\end{align} \\ -----------------\)