how do factor 3x^2 - 2x -6

Question

how do you factor 3x^2 - 2x -6.

Answer 1

Completing the square Method :

The expression is 3x² - 2x - 6.

Take out common factor.

= 3(x² - 2x/3 - 2)

To change the expression (x² - 2x/3 - 6) into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

 Here x coefficient = 2/3. So, (half the x coefficient)² = (1/3)² = 1/9.

Add and subtract 1/9 to the expression.

= 3(x² - 2x/3 + 1/9 - 1/9 - 2)

= 3[{x² - 2(1/3)x + (1/3)²} - (1 + 18)/9]

= 3[{x² - 2(1/3)x + (1/3)²} - 19/9]

Apply Perfect Square Trinomial : u² + 2uv + v² = (u + v)².

= 3[(x - 1/3)² - (√19/3)²]

Apply Difference of Two Squares : u² - v² = (u + v)(u - v).

= 3(x - 1/3 + √19/3)(x - 1/3 - √19/3)

= 3(x - 1/3 + √19/3)(x - 1/3 - √19/3)

= (1/3)(3x - 1 + √19)(3x - 1 - √19)

The factorization form of the expression is (1/3)(3x - 1 + √19)(3x - 1 - √19).