For What Values of M Does the Graph of Y = 3X2 + 7X + M Have Two X-Intercepts?
When analyzing the equation y = 3x^2 + 7x + m to find the values of m that result in two x-intercepts, we focus on the discriminant of the quadratic formula, b^2 – 4ac. For the given equation, the discriminant must be greater than zero for the graph to have two x-intercepts.
By setting up the inequality 7^2 – 4(3)(m) > 0 and solving for m, we can determine the specific values that result in this condition. This calculation allows us to identify the range of m values that produce two x-intercepts on the graph of y = 3x^2 + 7x + m.
Explaining X-intercepts
The X-intercepts of a quadratic equation correspond to the values of X where the graph of the equation intersects the X-axis. In other words, they are the points on the graph where the value of Y is equal to zero. The X-intercepts can be found by setting the equation equal to zero and solving for X. For the given equation Y = 3X^2 + 7X + M, we can find the X-intercepts by setting Y equal to zero:
| Equation | Simplified Equation |
|---|---|
| 3X^2 + 7X + M = 0 | |
By solving the simplified equation, we can determine the values of X where the graph of Y = 3X^2 + 7X + M will have two X-intercepts.
Graphing Quadratic Equations
When graphing quadratic equations like y = 3x^2 + 7x + M, it is important to understand the basics of graphing quadratic equations in order to identify the values of M that result in the graph having two x-intercepts. To plot the graph, we can use the standard form of a quadratic equation, which is y = ax^2 + bx + c.
In this case, with a = 3, b = 7, and c = M, we can substitute these values into the quadratic equation, y = 3x^2 + 7x + M, and solve for the x-intercepts by setting y = 0.
By using the quadratic formula or factoring, we can calculate the values of x that make y equal to zero, which will determine if the graph has two x-intercepts. If there are two distinct solutions, then the graph will have two x-intercepts. The values of M that satisfy this condition can be determined through algebraic calculations.
Conditions For Two X-intercepts
The graph of y = 3x2 + 7x + m has two x-intercepts when the discriminant, Δ, is greater than 0. The discriminant is calculated using the formula: Δ = b2 – 4ac, where a = 3, b = 7, and c = m. The conditions for two x-intercepts are fulfilled when Δ > 0. This means that 4ac – b2 should be less than 0.
Determining Values Of M
In the equation Y = 3X2 + 7X + M, the graph will have two x-intercepts if the discriminant is greater than zero. Using the discriminant formula, D = b2 – 4ac, we can determine the values of M. When the discriminant is greater than zero, the equation will have two real roots, leading to two x-intercepts. By solving the inequality D > 0, we can find the range of values for M that result in two x-intercepts for the given quadratic equation. Therefore, the values of M for which the graph of Y = 3X2 + 7X + M has two x-intercepts can be determined using the discriminant and solving the inequality D > 0.
Conclusion
To determine the values of M for which the graph of Y = 3X^2 + 7X + M has two x-intercepts, we analyzed the discriminant of the quadratic equation. By setting the discriminant greater than zero, we found that the graph will have two x-intercepts when 49 – 12M is greater than zero.
Therefore, for values of M where M is less than 4. 08, the equation will yield two x-intercepts.