15. Which of the following is equivalent to : \[ x^{2}-6 x+8 \] a. \( (x-2)(x-4) \) b. \( (x+2)(x-4) \) c. \( (x+2)(x+4) \) d. \( (x+2)(x-4) \)

To factor the quadratic expression \( x^2 - 6x + 8 \), we look for two numbers that multiply to \( 8 \) (the constant term) and add up to \( -6 \) (the coefficient of \(x\)). The numbers \( -2 \) and \( -4 \) meet these criteria. Thus, the expression can be factored as \( (x-2)(x-4) \). To confirm our factors, we can expand \( (x-2)(x-4) \) back out: \[ (x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8. \] So, the correct choice is (a) \( (x-2)(x-4) \)! Keep practicing your factoring skills; they’ll come in handy in many areas of mathematics! Understanding factoring can also be a game changer when solving quadratic equations. If you find an expression you want to zero, like \( x^2 - 6x + 8 = 0 \), you can set \( (x-2)(x-4) = 0 \) and quickly find solutions at \( x = 2 \) and \( x = 4 \). Factoring can save you time compared to using the quadratic formula every time!