Question

Factor the following Trinomial:
x^2-3x+1

Answer 1

(x-3+√5)(x-3-√5)

Step-by-step explanation:

Use the quadratic formula and plug in the numbers.

By using the trinomial given, you should get (3±√5)/2

Then, plug into the factored form of a quadratic equation, which is (x-x₁)(x-x₂)

so we get (x-3+√5)(x-3-√5).

I hope this helped! Have a good day :)

Answer 2

The trinomial x^2-3x+1 does not factor nicely with rational coefficients. The quadratic formula would provide the roots, but they likely will not be rational numbers. Since no integers satisfy the conditions for factoring, we cannot factor this trinomial over the rationals.

Step-by-step explanation:

To factor the trinomial x^2-3x+1, we are looking for two binomials that multiply together to give the original trinomial. However, this trinomial does not factor nicely with rational coefficients. In such cases, one can use the quadratic formula which is applicable for any quadratic equation of the form ax^2 + bx + c = 0. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation x^2-3x+1=0, a=1, b=-3, and c=1. Applying these values to the quadratic formula, we would find the roots of the equation which represent the solutions but do not result in simple factors for the trinomial.

If we attempt to factor using factoring techniques, we look for two numbers that multiply to ac (which is 1) and add up to b (which is -3). As no two such integers exist, factoring over the rationals is not possible for this trinomial. The solutions could include irrational or complex numbers depending on the discriminant (b^2-4ac).