Problem 31 - Algebra and Trigonometry

Question

Factor each trinomial, or state that the trinomial is prime. $$9 x^{2}-9 x+2$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(9x^{2} - 9x + 2\) is \(9(x - 1)(9x - 2)\).

Step 1: Identify the coefficients in the trinomial

For the trinomial \(9x^{2} - 9x + 2\), the coefficients are a=9, b=-9, and c=2.

Step 2: Calculate the discriminant

Calculate the discriminant (\(D\)) using the formula \(D = b^{2} - 4ac\), so \(D = (-9)^{2} - 4*9*2 = 81 - 72 = 9\). The discriminant is positive, which means the equation has two real roots and can be factorized.

Step 3: Find the roots

Use the formula to find roots of a quadratic equation, \(x = \frac{-b ± \sqrt{D}}{2a}\). Substituting the values, \(x1 = \frac{-(-9) + \sqrt{9}}{2*9} = 1\) and \(x2 = \frac{-(-9) - \sqrt{9}}{2*9} = \frac{2}{9}\)

Step 4: Factorize the trinomial

Using the roots to factorize the trinomial, we get \(9x^{2} - 9x + 2 = 9(x -x1)(x -x2) = 9(x - 1)(9x - 2)\).