Certainly! Let's solve the problem step by step. The goal is to rewrite the equation with a positive coefficient for [tex]\( x^2 \)[/tex] on the left side and zero on the right side.
Given the inequality:
[tex]\[ 3x^2 - 5x + 7 < 2x^2 - 3x + 8 \][/tex]
First, let's move all the terms to one side of the inequality so that the right side becomes zero. We subtract the expression on the right from the expression on the left:
[tex]\[ 3x^2 - 5x + 7 - (2x^2 - 3x + 8) < 0 \][/tex]
Now, distribute the negative sign through the parentheses:
[tex]\[ 3x^2 - 5x + 7 - 2x^2 + 3x - 8 < 0 \][/tex]
Next, combine like terms:
[tex]\[ (3x^2 - 2x^2) + (-5x + 3x) + (7 - 8) < 0 \][/tex]
Simplify the terms inside the parentheses:
[tex]\[ x^2 - 2x - 1 < 0 \][/tex]
So the simplified form of the inequality, with the coefficient of [tex]\( x^2 \)[/tex] being positive and the right side being zero, is:
[tex]\[ x^2 - 2x - 1 < 0 \][/tex]
