Question

#15
$x^2 - 5x + 6$
$a\cdot c = 1\cdot 6$
$a = 1,$
$a = -5$
$c = 6$

Explanation:

Step1: Identify the quadratic form

The quadratic expression is \(x^{2}-5x + 6\), in the form \(ax^{2}+bx + c\) where \(a = 1\), \(b=- 5\), \(c = 6\). We need to factor it by finding two numbers that multiply to \(a\times c=1\times6 = 6\) and add up to \(b=-5\).

Step2: Find the two numbers

We need two numbers \(m\) and \(n\) such that \(m\times n=6\) and \(m + n=-5\). The numbers are \(-2\) and \(-3\) since \((-2)\times(-3)=6\) and \((-2)+(-3)=-5\).

Step3: Rewrite the middle term

Rewrite \(-5x\) as \(-2x-3x\), so the expression becomes \(x^{2}-2x-3x + 6\).

Step4: Group and factor

Group the first two terms and the last two terms: \((x^{2}-2x)+(-3x + 6)\). Factor out the common factors from each group: \(x(x - 2)-3(x - 2)\). Then factor out \((x - 2)\) to get \((x - 2)(x - 3)\).

Answer:

The factored form of \(x^{2}-5x + 6\) is \((x - 2)(x - 3)\)