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

Solution:

The given equation is of a polynomial which when represented on the graph represents a parabola. It is known that when the parabola crosses the x-axis the value of the variable “y” at the point of intersection with the x-axis is “zero”. Therefore substituting the value of “y” as zero in the given equation we have:

0 = 3x² + 7x + m

Now to factorize the above quadratic equation we have to assume the value of m. The possible value of “m” can be deduced with the help of the other two terms of the equation i.e. 3x2 and 7x. The appropriate value of m would therefore be “2”. Substituting the value of m we can factorize the above equation as follows.

0 = 3x² + 7x + 2

0 = (x + 2)(3x + 1)

m = 2 is a unique value as there is no other value of m which can factorize the given equation. Solving the above equation we get two values of x where y = 0 and these are x = -2 and x = -1/3. Hence for m = 2 the equation will have two values of intercepts.

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

Summary:

For m = 2 the parabola intersects the x-axis at two points (-2, 0) and (-1/3, 0).