For What Values of M Does the Graph of Y = 3X2 + 7X + M Have Two X-Intercepts?
For the graph of y = 3x^2 + 7x + M to have two x-intercepts, the discriminant (b^2 – 4ac) must be greater than 0, where a = 3, b = 7, and c = M. Therefore, M values for which the discriminant is positive will result in the graph having two x-intercepts. When analyzing the graph of a quadratic equation, it’s essential to consider the discriminant to determine the number of x-intercepts.
In this case, for the equation y = 3x^2 + 7x + M, having two x-intercepts implies that the quadratic equation must have real and distinct roots. By understanding the conditions for the discriminant to be greater than 0, we can identify the specific M values that satisfy this requirement, resulting in the graph having two x-intercepts.
Let’s delve deeper into the relationship between the discriminant and the M values for which the graph of the quadratic equation exhibits this behavior.
Determination Of X-intercepts
When determining the values of M that result in the graph of y = 3x^2 + 7x + M having two x-intercepts, the discriminant formula is used. The discriminant formula, Δ = b^2 – 4ac, where a, b, and c are the coefficients of the quadratic equation, helps identify the nature of the roots. For two x-intercepts to exist, the discriminant must be greater than zero.
This condition ensures that the equation has two distinct real solutions. By substituting the values of a, b, and c from y = 3x^2 + 7x + M into the discriminant formula, we can determine the range of values for M that satisfy this condition. If the discriminant is greater than zero, the equation has two x-intercepts. If the discriminant is less than zero, the equation has complex roots. If the discriminant is equal to zero, the equation has one x-intercept.
In summary, to find the values of M that yield two x-intercepts for the equation y = 3x^2 + 7x + M, we utilize the discriminant formula, setting it greater than zero. Solving this inequality equation will provide the range of M values that result in two x-intercepts.
Finding The Values Of M
In order to find the values of M for which the graph of y = 3x^2 + 7x + M has two x-intercepts, we need to analyze the discriminant of the quadratic equation. The discriminant is the part of the quadratic formula that helps determine the nature and number of solutions. For a quadratic equation in the form of ax^2 + bx + c = 0, the discriminant Δ is calculated as b^2 – 4ac.
If the discriminant is greater than zero, the equation has two distinct real solutions, resulting in two x-intercepts. If the discriminant is equal to zero, the equation has one real solution and one repeating solution, resulting in one x-intercept. If the discriminant is less than zero, the equation has two complex solutions, meaning there are no x-intercepts. By setting the discriminant greater than zero, we can solve for the values of M that satisfy this condition.
Conclusion
Based on the analysis, the values of M for which the graph of y = 3x^2 + 7x + M has two x-intercepts can be determined. The discriminant, calculated as b^2 – 4ac, plays a crucial role in this determination.
If the discriminant is positive, the equation has two distinct real roots and, thus, two x-intercepts. However, if the discriminant is negative or zero, there will be no or only one x-intercept. This understanding allows for a clear interpretation of the graph and its behavior within the given range.