Problem 22 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$x^{2}-14 x+45$$

Step-by-Step Solution

Answer

The trinomial \(x^{2}-14x+45\) factorises to \((x+9)(x+5)\).

Step 1: Recognizing the Structure

The given trinomial has the general structure of a quadratic equation, written in the form of \(ax^{2}+bx+c\) where \(a = 1, b = -14\), and \(c = 45\). We aim to factorise this equation into two binomials of the form \((x-p)(x-q)\), where \(p\) and \(q\) are the roots of the equation.

Step 2: Finding Values

We need to find two numbers that multiply to \(a*c = 1*45 = 45\) and add up to \(b = -14\). The numbers that satisfy these conditions are \(-9\) and \(-5\) since \(-9*-5 = 45\) and \(-9 - 5 = -14\).

Step 3: Writing Factorized Form

We substitute \(p\) and \(q\) with \(-9\) and \(-5\) respectively. Thus, our factorised form of the equation becomes: \((x+9)(x+5)\)