Example 7.20 from Elementary Algebra, 7.2 Factor Quadratic Trinomials with Leading Coefficient 1

Factor Quadratic Trinomials with Leading Coefficient 1: example 7.20

Factor: $t^{2} - 11 t + 28$ .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Again, with the positive last term, 28, and the negative middle term, $−11 t$ , we need two negative factors. Find two numbers that multiply 28 and add to $−11$ .

$\begin{matrix} t^{2} - 11 t + 28 \\ Write the factors as two binomials with first terms t . & (t) (t) \end{matrix}$

Find two numbers that: multiply to 28 and add to $−4 + (−7) = −11 *$ .

Factors of $28$ Sum of factors

$−1, −28$ $−1 + (−28) = −29$

$−2, −14$ $−2 + (−14) = −16$

$−4, −7$ $−4 + (−7) = −11 *$

$\begin{matrix} Use −4, −7 as the last terms of the binomials. & (t - 4) (t - 7) \\ Check. \\ (t - 4) (t - 7) \\ t^{2} - 7 t - 4 t + 28 \\ t^{2} - 11 t + 28 ✓ \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Factor: $u^{2} - 9 u + 18$ .

How did it go?