Use the discriminant to decide whether the expression can be factored?

Question

If it can be factored, factor the expression..

Guys and Gals, I seriously do need help.

The problems are these and can you please explain it step by step.

2x^2 - 3x - 2

3y^2 - 4y + 7

6x^2 - 4x - 5

for the first one, the answer is (2x + 1)(x-2)

but when i do that b^2-4ac, the answers are totally different

And thank you so much to the first two. I would definitely give you a thumbs up but i am not up to rate 2

Answer 1

Discriminant = b^2-4ac in this case. You can factor if the discriminant is a perfect square.

1) 9+16 is a perfect square, 25, so it can be factored.
(2x+1)(x-2)
2) 16 - 84 is negative, so it cannot be factored.

3) 16+120 = 136 is not a perfect square, so it cannot be factored

Answer 2

2x^2 - 3x - 2
then b^2 - 4ac = 9 - 4(2)(-2) = 9 + 16 = 25 which a square of 5
factor is (2x +1 )( x -2)

3y^2 -4y + 7
b^2 -4ac = 16 -4(3)(7) = 16 - 84= - -68, therefore no real factor

6X^2 -4x -5
b^2 - 4ac = 16 - 4(6)(-5) = 16 + 120 = 136, therefore with out factor in whole number.

Answer 3

The discriminant is b^2 -4 a c, so in the first example it is 9 - 4(2)2, which is -7, which is less than 0, so the expression has complex factors only. Do the others the same way; if the discriminant is zero, the expression has two identical factors; if it is positive, it has two different factors, and if it is a perfect square, it has rational factors. It is customary to say that if there are not rational factors, the expression is prime.

Answer 4

discriminant = b^2 - 4ac.
If it comes out positive and a perfect square, then the expression can be factored with rational numbers.

2x^2 - 3x - 2
b^2 - 4ac = (-3)^2 - 4(2)(-2) = 9 + 16 = 25 pos and perfect square

2x^2 - 3x - 2 = ( 2x + 1)(x - 2)
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3y^2 - 4y + 7
b^2 - 4ac = (-4)^2 - 4(3)(7) = 16 - 84 = - 68 negative and not perfect square, so not factorable

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6x^2 - 4x - 5
b^2 - 4ac = (-4)^2 - 4(6)(-5) = 16 + 120 = 136 positive but not a perfect square.
cannot be factored with rational numbers

Answer 5

The discriminant is a fancy word for the stuff under the square root sign in the quadratic formula (b^2-4ac).

As long as b^2-4ac is a perfect square (and not negative), it will factor.

I think the first is 25 (so it is factorable), the second is negative so it certainly isn't factorable, and the third isn't a perfect square (136).