Question

Answer 1

a, \(x\left(x-2\right)+x-2=0\)

\(\Leftrightarrow\left(x-2\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-1\end{matrix}\right.\)

b, ​​\(5x\left(x-3\right)-x+3=0\)

\(\Leftrightarrow\left(x-3\right)\left(5x-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\\x=\dfrac{1}{5}\end{matrix}\right.\)

Answer 2

c, \(x^2\left(x-2011\right)-x+2011=0\)

\(\Leftrightarrow\left(x-2011\right)\left(x^2-1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2011\\x^2=1\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2011\\x=\pm1\end{matrix}\right.\)

Answer 3

a. x(x - 2) + x - 2 = 0

<=> (x + 1)(x - 2) = 0

<=> \(\left[{}\begin{matrix}x+1=0\\x-2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=2\end{matrix}\right.\)

b. 5x(x - 3) - x + 3 = 0

<=> 5x(x - 3) - (x - 3) = 0

<=> (5x - 1)(x - 3) = 0

<=> \(\left[{}\begin{matrix}5x-1=0\\x-3=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{1}{5}\\x=3\end{matrix}\right.\)

c. x²(x - 2011) - x + 2011 = 0

<=> x²(x - 2011) - (x - 2011) = 0

<=> (x² - 1)(x - 2011) = 0

<=> \(\left[{}\begin{matrix}x^2-1=0\\x-2011=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left(x-1\right)\left(x+1\right)=0\\x=2011\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x-1=0\\x+1=0\\x=2011\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-1\\x=2011\end{matrix}\right.\)

Answer 4

d, \(4x^2-36x+56=0\)

\(\Leftrightarrow4x^2-8x-28x+56=0\)

\(\Leftrightarrow4x\left(x-2\right)-28\left(x-2\right)=0\)

\(\Leftrightarrow4\left(x-2\right)\left(x-7\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=7\end{matrix}\right.\)

Answer 5

e, \(x^3-7x-6=0\)

\(\Leftrightarrow x^3-x-6x-6=0\)

\(\Leftrightarrow\left(x^2-x\right)\left(x+1\right)-6\left(x+1\right)=0\)

\(\Leftrightarrow\left(x^2-x-6\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left(x+2\right)\left(x-3\right)\left(x+1\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=-2\\x=3\\x=-1\end{matrix}\right.\)

Answer 6

f, \(\left(x+4\right)^2-\left(x+1\right)\left(x-1\right)=16\)

\(\Leftrightarrow x^2+8x+16-x^2+1=16\)

\(\Leftrightarrow8x+1=0\)

\(\Leftrightarrow x=-\dfrac{1}{8}\)

Answer 7

g, \(\left(x^2+x\right)\left(x^2+x+1\right)=6\)

\(\Leftrightarrow\left(x^2+x\right)^2+\left(x^2+x\right)-6=0\)

\(\Leftrightarrow\left(x^2+x-2\right)\left(x^2+x+3\right)=0\)

\(\Leftrightarrow x^2+x-2=0\)

\(\Leftrightarrow\left(x-1\right)\left(x+2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-2\end{matrix}\right.\)

Answer 8

f, \(x\left(x+1\right)\left(x-1\right)\left(x+2\right)=24\)

\(\Leftrightarrow\left(x^2+x\right)\left(x^2+x-2\right)=24\)

\(\Leftrightarrow\left(x^2+x\right)^2-2\left(x^2+x\right)-24=0\)

\(\Leftrightarrow\left(x^2+x\right)^2-6\left(x^2+x\right)+4\left(x^2+x\right)-24=0\)

\(\Leftrightarrow\left(x^2+x-6\right)\left(x^2+x+4\right)=0\)

\(\Leftrightarrow x^2+x-6=0\)

\(\Leftrightarrow\left(x-2\right)\left(x+3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=-3\end{matrix}\right.\)

Answer 9

d. 4x² - 36x + 56 = 0

<=> 4x² - 8x - 28x + 56 = 0

<=> 4x(x - 2) - 28(x - 2) = 0

<=> (4x - 28)(x - 2) = 0

<=> \(\left[{}\begin{matrix}4x-28=0\\x-2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=7\\x=2\end{matrix}\right.\)

e. x³ - 7x - 6 = 0

<=> x³ + 1 - 7x - 7 = 0

<=> (x + 1)(x² - x + 1) - 7(x + 1) = 0

<=> (x + 1)(x² - x + 1 - 7) = 0

<=> (x + 1)(x² - x - 6) = 0

<=> (x + 1)(x² + 2x - 3x - 6) = 0

<=> (x + 1)\(\left[x\left(x+2\right)-3\left(x+2\right)\right]\) = 0

<=> (x + 1)(x - 3)(x + 2) = 0

<=> \(\left[{}\begin{matrix}x+1=0\\x-3=0\\x+2=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=3\\x=-2\end{matrix}\right.\)

f. (x + 4)² - (x + 1)(x - 1) = 16

<=> (x + 4)² - x² + 1 - 4² = 0

<=> 4(2x + 4) - 15 = 0

<=> 8x + 16 - 15 = 0

<=> 8x + 1 = 0

<=> 8x = -1

<=> x = \(\dfrac{-1}{8}\)