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How to factor x² − 5x + 6

The product-sum method, a direct check, and the connection to polynomial roots.

$x^2-5x+6=(x-2)(x-3)$ because $(-2)(-3)=6$ and $-2+(-3)=-5$ .

The product-sum method

For a monic quadratic $x^2+bx+c$ , find numbers $r$ and $s$ with $r+s=b$ and $rs=c$ . Here their sum must be $-5$ and product $6$ , so they are $-2$ and $-3$ .

Check

Expanding gives $(x-2)(x-3)=x^2-3x-2x+6=x^2-5x+6.$ This quick check catches sign errors.

Connection to solutions

From $(x-2)(x-3)=0$ , either $x=2$ or $x=3$ . The roots are exactly the values that make the polynomial zero.

Frequently asked questions

What if no integer pair works?

Use the quadratic formula to find the roots, then write the corresponding factors when the roots lie in your chosen number system.

Why do factor signs oppose the roots?

Because the factor that becomes zero at x=r is x-r.