When graphing a quadratic equation, the shape of the graph can tell you a lot about the equation itself. Specifically, the number of x-intercepts, or points where the graph intersects the x-axis, can be determined by the equation’s coefficients and constant. This article will discuss for what values of m does the graph y = 3×2 + 7x + m have two x-intercepts.
Determining M Values
In order for a graph to have two x-intercepts, the equation must have two solutions. To determine the values of m for which the equation has two solutions, we must use the Quadratic Formula. The Quadratic Formula states that the solutions of the equation ax2 + bx + c = 0 are given by:
x = [-b ± √(b2 – 4ac)] / 2a
In this equation, a = 3, b = 7, and c = m. Thus, the solutions are given by:
x = [-7 ± √(72 – 4(3)(m))] / 2(3)
Since we are looking for two solutions, the expression under the square root must be equal to or greater than zero. Thus, we can set the expression equal to zero and solve for m:
72 – 4(3)(m) = 0
72 – 12m = 0
12m = 72
m = 6
Therefore, for the graph of y = 3×2 + 7x + m to have two x-intercepts, the value of m must be 6.
Two X-intercepts for Y = 3×2 + 7x + M
Now that we have determined the value of m for which the equation has two x-intercepts, we can use the Quadratic Formula to solve for the x-intercepts. Plugging m = 6 into the equation, we get:
x = [-7 ± √(72 – 4(3)(6))] / 2(3)
x = [-7 ± √(0)] / 2(3)
x = [-7 ± 0] / 2(3)
x = [-7 / 6] = -1.17, -5.83
Thus, for the graph of y = 3×2 + 7x + 6, the x-intercepts are -1.17 and -5.83.