Question

For what values of m does the graph of y = 3x^2 + 7x + m have two x-intercepts?
a) m>12/49
b) m<12/49
c) m<49/12
d) m> 49/12

Answer 1

The quadratic formula, has a part we call the "discriminant" defined by the variables that are inside the square root, and is denotated by "delta":
\[\Delta = b^2 - 4ac\]
Whenever we solve a quadratic equation that is complete and we analyze the discriminant, we can get 3 scenarios:
\[if \rightarrow \Delta > 0 =>\exists x_1, x_2 / ax^2 +bx+c=0\]
This just means: "if the discriminant is greater than zero, there will exist two x-intercepts"
And for the second scenario:
\[if \rightarrow \Delta = 0 \rightarrow \exists x_o / ax^2 + bx + c=0\]
This means: "if the discriminant is equal to zero, there will be one and only one x-intercept"
And for the last scenario:
\[if \rightarrow \Delta < 0 \rightarrow \exists x \notin \mathbb{R} / ax^2 + bx + c=0\]
This means that :"if the discriminant is less than zero, there will be no x-intercepts"
So, if we take your excercise and analyze the the discriminant:
\[3x^2 + 7x+m = y\]
we will find the values that satisfy y=0 :
\[3x^2 + 7x +m =0\]
And we'll analyze the discriminant:
\[\Delta=7^2 - 4(3)(m)\]
And we are only interested in the values that make the discriminant equal zero:
\[7^2 - 4(3)(m)=0\]
All you have to do is solve for "m".