\( 2 x ^ { 2 } - 5 x + 3 = ( 2 x - 3 ) ( x - \cdots \cdots \cdots ) \)

To solve the equation \( 2 x^{2} - 5 x + 3 = ( 2 x - 3 ) ( x - \cdots \cdots \cdots ) \), we need to find the missing factor in \( ( x - \cdots \cdots \cdots ) \). We can start by expanding the right side. By distributing, we get: \[ (2x - 3)(x - r) = 2x^2 - 2rx - 3x + 3r \] Grouping the terms gives us: \[ 2x^2 + (-2r - 3)x + 3r \] For this to equal \( 2x^{2} - 5x + 3 \), we match coefficients. That is: 1. From the \(x\) terms: \(-2r - 3 = -5\) 2. From the constant terms: \(3r = 3\) From the second equation, \( r = 1 \). Substituting \(r\) into the first equation: \[ -2(1) - 3 = -5 \] This holds true. So, we can conclude: \[ 2 x^{2} - 5 x + 3 = ( 2 x - 3 )( x - 1 ) \] Now we solved for the expression, we found \( (x - 1) \) fills the gap! Doing this can remind you of how important factoring is in algebra. It’s like finding the right puzzle piece to complete the image! Factoring allows you to simplify complex expressions and even solve equations with ease. Plus, mastering this skill makes higher-level math a lot more manageable. Keep practicing and you’ll be a factoring pro in no time!