For what values of m does the graph of y = 3x2 + 7x + m have two x-intercepts?

The answer is C. m < 49/12

So we see the leading coefient is positive (the 3 in the 3x^2 is positive) so the paraobola opens upward

so for it to have 2 x intercepts, we must have the vertex below the x axis

we must have the y value of the vertex negative

so

a hack is this

to move a function up c units, add c to whole function

also

for

y=ax^2+bx+c

the x value of the vertex is - b/2a

the y value is found by subsituting that value for x

so

y=3x^2+7x+m

x value of vertex is

-7 / (2*3) = - 7/6

if we sub it in

y=3 (-7/6) ^2+7 (-7/6) + m

y=49/12-49/6+m

we want the y value to be less than 0

so

0>49/12-49/6+m

0>49/12-98/12+m

0>-49/12+m

49/12>m

so it will have 2 x intercept for all m such that m<49/12