Question

which expression is prime? a $x^2 - 4x + 3$ b $x^2 - 6x + 9$ c $3x^2 + 5x - 2$ d $6x^2 - 7x + 18$

Explanation:

Step1: Factor Option A

For \(x^2 - 4x + 3\), find two numbers that multiply to \(3\) and add to \(-4\). The numbers are \(-1\) and \(-3\). So, \(x^2 - 4x + 3=(x - 1)(x - 3)\).

Step2: Factor Option B

For \(x^2 - 6x + 9\), it is a perfect square trinomial. \(x^2 - 6x + 9=(x - 3)^2\).

Step3: Factor Option C

For \(3x^2 + 5x - 2\), use the AC method. \(AC = 3\times(-2)=-6\). Find two numbers that multiply to \(-6\) and add to \(5\) (which are \(6\) and \(-1\)). Rewrite the middle term: \(3x^2 + 6x - x - 2 = 3x(x + 2)-1(x + 2)=(3x - 1)(x + 2)\).

Step4: Check Option D

For \(6x^2 - 7x + 18\), calculate the discriminant \(D = b^2 - 4ac=(-7)^2 - 4\times6\times18 = 49 - 432=-383\). Since the discriminant is negative, it cannot be factored over the real numbers, so it is prime.

Answer:

D. \(6x^2 - 7x + 18\)