\(x^2-x+1=\left(x^2-x+\dfrac{1}{4}\right)+\dfrac{3}{4}=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}>0\)

Bài 1. Rút gọn:

\(a, x\left(1-x\right)+6\left(x+3\right)\left(x+3\right)\)

\(=x-x^2+6\left(x^2+6x+9\right)\)

\(=x-x^2+6x^2+36x+54\)

\(=5x^2+37x+54\)

\(b, \left(2-3x\right)\left(2+3x\right)-\left(x+5\right)\left(x-5\right)\)

\(=\left(4-9x^2\right)-\left(x^2-25\right)\)

\(=-10x^2+29\)

\(c, \left(3x+1\right)\left(x+5\right)-\left(x-1\right)\left(x+1\right)\)

\(=3x^2+15x+x+5-x^2+1\)

\(=2x^2+16x+6\)

\(d,\left(2-3x\right)\left(2x+3\right)+6\left(x-1\right)^2\)

\(=\left(4x+6-6x^2-9x\right)+6\left(x^2-2x+1\right)\)

\(=4x+6-6x^2-9x+6x^2-12x+6\)

\(=-17x+12\)

\(e, x\left(5-x\right)-\left(2x+2\right)\left(3x+2\right)-\left(x-2\right)\left(x+2\right)\)

\(=5x-x^2-\left(6x^2+4x+6x+4\right)-\left(x^2-4\right)\)

\(=5x-x^2-6x^2-4x-6x-4-x^2+4\)

\(=-8x^2-5x\)

\(2^{n-1}.10=\left(2^n:2\right)\left(2.5\right)=2^2.5\)

\(2x^2+5x-3=0\)

\(\Leftrightarrow2x\left(x+3\right)-\left(x+3\right)=0\)

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

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

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

a) \(2x^2+5x-18\)

\(=2x^2-4x+9x-18\)

\(=2x\left(x-2\right)+9\left(x-2\right)\)

\(=\left(x-2\right)\left(2x+9\right)\)

b) \(4x^2-17x+15\)

\(=4x^2-12x-5x+15\)

\(=4x\left(x-3\right)-5\left(x-3\right)\)

\(=\left(x-3\right)\left(4x-5\right)\)

c) \(-8x^2+10x+7\)

\(=-8x^2-4x+14x+7\)

\(=-4x\left(2x+1\right)+7\left(2x+1\right)\)

\(=\left(2x+1\right)\left(-4x+7\right)\)

d) \(7x^2-30x+8\)

\(=7x^2-28x-2x+8\)

\(=7x\left(x-4\right)-2\left(x-4\right)\)

\(=\left(x-4\right)\left(7x-2\right)\)

e) \(-x^3+11x^2-30x\)

\(=x\left(-x^2+11x-30\right)\)

\(=x\left(-x^2+5x+6x-30\right)\)

\(=x\left[-x\left(x-5\right)+6\left(x-5\right)\right]\)

\(=x\left(x-5\right)\left(-x+6\right)\)

7) \(5x^2-3x-2\)

\(=5x^2-5x+2x-2\)

\(=5x\left(x-1\right)+2\left(x-1\right)\)

\(=\left(x-1\right)\left(5x+2\right)\)

8) \(x^2-4x-1\)

\(=\left(x^2-4x+4\right)-5\)

\(=\left(x-2\right)^2-5\)

\(=\left(x-2+\sqrt{5}\right)\left(x-2-\sqrt{5}\right)\)

9) \(x^2-7xy+12y^2\)

\(=x^2-3xy-4xy+12y^2\)

\(=x\left(x-3y\right)-4y\left(x-3y\right)\)

\(=\left(x-3y\right)\left(x-4y\right)\)

4) \(x^2+5x+4\)

\(=x^2+x+4x+4\)

\(=x\left(x+1\right)+4\left(x+1\right)\)

\(=\left(x+1\right)\left(x+4\right)\)

5) \(4x^2+4x-8\)

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

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

\(=4\left[x\left(x-1\right)+2\left(x-1\right)\right]\)

\(=4\left(x-1\right)\left(x+2\right)\)

6) \(x^2+x-42\)

\(=x^2-6x+7x-42\)

\(=x\left(x-6\right)+7\left(x-6\right)\)

\(=\left(x-6\right)\left(x+7\right)\)

1) \(x^4-3x^2-x+3\)

\(=x\left(x^3-1\right)-3\left(x^2-1\right)\)

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

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

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

2) \(3x+3y-x^2-2xy-y^2\)

\(=3\left(x+y\right)-\left(x^2+2xy+y^2\right)\)

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

\(=\left(x+y\right)\left(3-x-y\right)\)

3) \(x^4-x\)

\(=x\left(x^3-1\right)\)

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

\(2x^2-5xy-3y^2\)

\(=2x^2-6xy+xy-3y^2\)

\(=2x\left(x-3y\right)+y\left(x-3y\right)\)

\(=\left(x-3y\right)\left(2x+y\right)\)

\(i, x^4+4x^2+16\)

\(=\left(x^4+8x^2+16\right)-4x^2\)

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

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