Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
- \(x(16x+16)=(x+1)\)
- \(6x^2-(12x-49)=2x(x-8)\)
- \(-(7-17x)=-x^2-(-11-14x)\)
- \(3x^2-(14x-16)=2x(x-2)\)
- \(\frac{1}{3}x^2-\frac{1}{3}x-10=0\)
- \(\frac{8}{3}x=-\frac{16}{33}x^2-\frac{11}{3}\)
- \(2x=-\frac{3}{7}x^2-\frac{7}{3}\)
- \(-(10-28x)=-x^2-(74-16x)\)
- \(x(36x+27)=2(x-2)\)
- \(x(x-41)=-30(x+1)\)
- \(x(x-20)=4(x-36)\)
- \(x(x+27)=11(x-5)\)
Gebruik de discriminant om volgende vierkantsvergelijkingen op te lossen
Verbetersleutel
- \(x(16x+16)=(x+1) \\
\Leftrightarrow 16x^2+16x=x+1 \\
\Leftrightarrow 16x^2+15x-1=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+15x-1=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (15)^2-4.16.(-1) & &\\
& = 225+64 & & \\
& = 289 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-15-\sqrt289}{2.16} & & = \frac{-15+\sqrt289}{2.16} \\
& = \frac{-32}{32} & & = \frac{2}{32} \\
& = -1 & & = \frac{1}{16} \\ \\ V &= \Big\{ -1 ; \frac{1}{16} \Big\} & &\end{align} \\ -----------------\)
- \(6x^2-(12x-49)=2x(x-8) \\
\Leftrightarrow 6x^2-12x+49=2x^2-16x \\
\Leftrightarrow 4x^2+4x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{4x^2+4x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (4)^2-4.4.49 & &\\
& = 16-784 & & \\
& = -768 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(-(7-17x)=-x^2-(-11-14x) \\
\Leftrightarrow -7+17x=-x^2+11+14x \\
\Leftrightarrow x^2+3x-18=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+3x-18=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (3)^2-4.1.(-18) & &\\
& = 9+72 & & \\
& = 81 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-3-\sqrt81}{2.1} & & = \frac{-3+\sqrt81}{2.1} \\
& = \frac{-12}{2} & & = \frac{6}{2} \\
& = -6 & & = 3 \\ \\ V &= \Big\{ -6 ; 3 \Big\} & &\end{align} \\ -----------------\)
- \(3x^2-(14x-16)=2x(x-2) \\
\Leftrightarrow 3x^2-14x+16=2x^2-4x \\
\Leftrightarrow x^2-10x+16=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-10x+16=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-10)^2-4.1.16 & &\\
& = 100-64 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-10)-\sqrt36}{2.1} & & = \frac{-(-10)+\sqrt36}{2.1} \\
& = \frac{4}{2} & & = \frac{16}{2} \\
& = 2 & & = 8 \\ \\ V &= \Big\{ 2 ; 8 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{1}{3}x^2-\frac{1}{3}x-10=0\\
\Leftrightarrow \color{red}{3.} \left(\frac{1}{3}x^2-\frac{1}{3}x-10\right)=0 \color{red}{.3} \\
\Leftrightarrow x^2-x-30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-x-30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-1)^2-4.1.(-30) & &\\
& = 1+120 & & \\
& = 121 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-1)-\sqrt121}{2.1} & & = \frac{-(-1)+\sqrt121}{2.1} \\
& = \frac{-10}{2} & & = \frac{12}{2} \\
& = -5 & & = 6 \\ \\ V &= \Big\{ -5 ; 6 \Big\} & &\end{align} \\ -----------------\)
- \(\frac{8}{3}x=-\frac{16}{33}x^2-\frac{11}{3} \\
\Leftrightarrow \frac{16}{33}x^2+\frac{8}{3}x+\frac{11}{3}=0 \\
\Leftrightarrow \color{red}{33.} \left(\frac{16}{33}x^2+\frac{8}{3}x+\frac{11}{3}\right)=0 \color{red}{.33} \\
\Leftrightarrow 16x^2+88x+121=0 \\\text{We zoeken de oplossingen van } \color{blue}{16x^2+88x+121=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (88)^2-4.16.121 & &\\
& = 7744-7744 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-88}{2.16} & & \\
& = -\frac{11}{4} & & \\V &= \Big\{ -\frac{11}{4} \Big\} & &\end{align} \\ -----------------\)
- \(2x=-\frac{3}{7}x^2-\frac{7}{3} \\
\Leftrightarrow \frac{3}{7}x^2+2x+\frac{7}{3}=0 \\
\Leftrightarrow \color{red}{21.} \left(\frac{3}{7}x^2+2x+\frac{7}{3}\right)=0 \color{red}{.21} \\
\Leftrightarrow 9x^2+42x+49=0 \\\text{We zoeken de oplossingen van } \color{blue}{9x^2+42x+49=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (42)^2-4.9.49 & &\\
& = 1764-1764 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-42}{2.9} & & \\
& = -\frac{7}{3} & & \\V &= \Big\{ -\frac{7}{3} \Big\} & &\end{align} \\ -----------------\)
- \(-(10-28x)=-x^2-(74-16x) \\
\Leftrightarrow -10+28x=-x^2-74+16x \\
\Leftrightarrow x^2+12x+64=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+12x+64=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (12)^2-4.1.64 & &\\
& = 144-256 & & \\
& = -112 & & \\ & < 0 \\V &= \varnothing \end{align} \\ -----------------\)
- \(x(36x+27)=2(x-2) \\
\Leftrightarrow 36x^2+27x=2x-4 \\
\Leftrightarrow 36x^2+25x+4=0 \\\text{We zoeken de oplossingen van } \color{blue}{36x^2+25x+4=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (25)^2-4.36.4 & &\\
& = 625-576 & & \\
& = 49 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-25-\sqrt49}{2.36} & & = \frac{-25+\sqrt49}{2.36} \\
& = \frac{-32}{72} & & = \frac{-18}{72} \\
& = \frac{-4}{9} & & = \frac{-1}{4} \\ \\ V &= \Big\{ \frac{-4}{9} ; \frac{-1}{4} \Big\} & &\end{align} \\ -----------------\)
- \(x(x-41)=-30(x+1) \\
\Leftrightarrow x^2-41x=-30x-30 \\
\Leftrightarrow x^2-11x+30=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-11x+30=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-11)^2-4.1.30 & &\\
& = 121-120 & & \\
& = 1 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-(-11)-\sqrt1}{2.1} & & = \frac{-(-11)+\sqrt1}{2.1} \\
& = \frac{10}{2} & & = \frac{12}{2} \\
& = 5 & & = 6 \\ \\ V &= \Big\{ 5 ; 6 \Big\} & &\end{align} \\ -----------------\)
- \(x(x-20)=4(x-36) \\
\Leftrightarrow x^2-20x=4x-144 \\
\Leftrightarrow x^2-24x+144=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2-24x+144=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (-24)^2-4.1.144 & &\\
& = 576-576 & & \\
& = 0 & & \\ x & = \frac{-b\pm \sqrt{D}}{2.a} & & \\
& = \frac{-(-24)}{2.1} & & \\
& = 12 & & \\V &= \Big\{ 12 \Big\} & &\end{align} \\ -----------------\)
- \(x(x+27)=11(x-5) \\
\Leftrightarrow x^2+27x=11x-55 \\
\Leftrightarrow x^2+16x+55=0 \\\text{We zoeken de oplossingen van } \color{blue}{x^2+16x+55=0} \\ \\\begin{align}
D & = b^2 - 4.a.c & & \\
& = (16)^2-4.1.55 & &\\
& = 256-220 & & \\
& = 36 & & \\ \\
x_1 & = \frac{-b-\sqrt{D}}{2.a} & x_2 & = \frac{-b+\sqrt{D}}{2.a} \\
& = \frac{-16-\sqrt36}{2.1} & & = \frac{-16+\sqrt36}{2.1} \\
& = \frac{-22}{2} & & = \frac{-10}{2} \\
& = -11 & & = -5 \\ \\ V &= \Big\{ -11 ; -5 \Big\} & &\end{align} \\ -----------------\)