Question

Решите уравнение |2x^2+5x-3|=2x-1

Answer 1

 

$|2x^2 + 5x - 3| = 2x - 1\\ 2x - 1 geq 0, 2x geq 1, x geq frac{1}{2}\\ 1) 2x^2 + 5x - 3 > 0\\ x_1*x_2 = frac{-3}{2} = -3 * frac{1}{2}\\ x_1 + x_2 = frac{-5}{2} = frac{1}{2} - 3\\ x_1 = -3, x_2 = frac{1}{2}\\ <var>x in (-infty,-3) cup (frac{1}{2}, +infty)</var>\\ 2x^2 + 5x - 3 = 2x - 1\\ 2x^2 + 3x - 2 = 0\\ x_1*x_2 = -1 = -2*frac{1}{2}\\ x_1 + x_2 = -frac{3}{2} = -2 + frac{1}{2}\\ x_1 = -2, x_2 = frac{1}{2}$

 

 

$2) 2x^2 + 5x - 3 leq 0\\ <var>x in [-3,frac{1}{2}]</var>\\ -2x^2 -5x + 3 = 2x - 1\\ -2x^2 - 7x + 4 = 0\\ x_1*x_2 = -2 = -4*frac{1}{2}\\ x_1 + x_2 = -3.5 = -4 + frac{1}{2}\\ x_1 = -4, <var>underline{</var>x_2 = frac{1}{2}}\\\\ fbox{x = 0.5}$