To determine the values of m for which the graph of the quadratic function y = 3x² + 7x + m has two x-intercepts, we need to analyze the discriminant of the quadratic equation. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are coefficients. In this case, we have:
- a = 3
- b = 7
- c = m
The discriminant D of a quadratic equation is given by the formula:
D = b² – 4ac
For the quadratic to have two distinct x-intercepts, the discriminant must be greater than zero:
D > 0
Substituting the values of a, b, and c into the discriminant:
D = 7² – 4(3)(m) = 49 – 12m
Now, we set the discriminant greater than zero:
49 – 12m > 0
To solve this inequality:
- Subtract 49 from both sides:
- Divide by -12, reversing the inequality sign:
-12m > -49
m < rac{49}{12}
Thus, the graph of y = 3x² + 7x + m will have two x-intercepts for values of m that are less than rac{49}{12} ext{ or approximately } 4.08.