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Chapter 5: Problem 37

Factor. $$t^{2}-9 t+14$$

Short Answer

Expert verified
The factored form of \(t^{2}-9 t+14\) is \((t - 2)(t - 7)\).

Step by step solution

01

Identify the coefficients and constant

From the given expression \(t^{2}-9 t+14\), the coefficients of 't' are 1 and -9, and the constant term is 14.
02

Find two numbers that add up to -9 and multiply to 14

To factor a quadratic, we need to find two numbers that add up to the coefficient of 't' (-9) and multiply to the constant term (14). There are several pairs of numbers that multiply to 14 (for example, 1 and 14, 2 and 7) but the correct pair here is -2 and -7, as they satisfy both conditions.
03

Rewrite the original expression

The quadratic expression \(t^{2}-9 t+14\) is then rewritten as: \(t^{2}-2 t-7 t+14\).
04

Factoring by grouping

By factoring by grouping, we pair the first two terms together and the last two terms together, then factor out a common factor from each binomial: \(t(t - 2) - 7(t - 2)\).
05

Factor out the common binomial

Finally, factor out the common binomial, giving you: \((t - 2)(t - 7)\).

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Most popular questions from this chapter

Explain how to determine whether a trinomial is prime.

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