Question

katie did not factor below correctly. what did she do wrong and what is the correct answer?
$x^2 - 6x + 8$
$4 + 2 = 6$
$4 * 2 = 8$
katies answer: $(x - 4)(x + 2)$

Explanation:

Step1: Recall factoring trinomials

To factor \(x^2 + bx + c\) (or \(x^2 - bx + c\) here), we need two numbers that multiply to \(c\) and add up to \(b\) (with sign considered). For \(x^2 - 6x + 8\), we need two numbers that multiply to \(8\) and add to \(-6\) (since the middle term is \(-6x\)).

Step2: Analyze Katie's mistake

Katie used \(4\) and \(2\), but in her factored form \((x - 4)(x + 2)\), when we expand it: \(x^2+2x - 4x - 8=x^2 - 2x - 8\), which is not the original trinomial. The error is in the sign of the second number. We need two negative numbers because their product is positive (\(8\)) and their sum is negative (\(-6\)). The correct numbers are \(-4\) and \(-2\) (since \(-4\times(-2) = 8\) and \(-4+(-2)=-6\)).

Step3: Correct factoring

Using the correct numbers, we factor \(x^2 - 6x + 8\) as \((x - 4)(x - 2)\). Let's verify: \((x - 4)(x - 2)=x^2-2x - 4x + 8=x^2 - 6x + 8\), which matches the original trinomial.

Answer:

Katie's mistake was using \(+2\) instead of \(-2\) in the factored form (the signs of the constants in the binomials were incorrect; she should have two negative constants since their product is positive and sum is negative). The correct answer is \((x - 4)(x - 2)\) (or also \((x - 2)(x - 4)\)).