Explain why we know that 3x^2+4x−2 is not factorable.
We know that 3x^2+4x−2 is not factorable because its discriminant is negative. The discriminant is the part of the quadratic formula under the square root sign, which is b^2-4ac. In this case, a=3, b=4, and c=-2. So, the discriminant is:
b^2-4ac = 4^2 - 4(3)(-2) = 16 + 24 = 40
Since the discriminant is positive, we know that the quadratic has two real roots. However, if the discriminant were negative, the quadratic would have two complex roots, which means it cannot be factored using real numbers. Therefore, since the discriminant is positive, we know that 3x^2+4x−2 is not factorable.
since the discriminant is not a perfect square, the roots cannot be rational.
Hence no factorization is possible.