Reviewed by Phoebe Belza-Barrientos
For what values of m does the graph of y = 3x^2 + 7x + m have two x-intercepts?
Answer: The graph of y = 3x^2 + 7x + m has two x-intercepts for values of m less than 49/12
The equation y = 3x^2 + 7x + m represents a quadratic function. The x-intercepts (real roots) of this quadratic occur where y = 0. For the graph to have two distinct x-intercepts, the quadratic equation must have two real solutions, which happens if its discriminant is positive.
Methods
Math Tutor Explanation Using the Discriminant Method
To determine for which values of m the quadratic equation has two x-intercepts, examine the discriminant from the quadratic formula.
Step 1: Write down the discriminant formula: D = b^2 - 4ac
Step 2: Substitute a = 3, b = 7, and c = m into the formula to get D = 7^2 - 4 × 3 × m
Math Tutor Explanation Using Analysis of the Parabola
Understand how the value of m affects the position of the graph and its intersection with the x-axis.
Step 1: Realize that the vertex moves up and down as m changes
Step 2: Two x-intercepts exist when the entire parabola crosses the x-axis twice, which depends on the value of m in relation to the discriminant being positive
Step 1:
Step 2:
Math Tutor suggests: More Practice with Quadratic Graphs and X-Intercepts
Deepen your understanding of quadratic equations, their graphs, and how to determine the number of x-intercepts by exploring these related exercises.
FAQ on Quadratic Equations and X-Intercepts
What is the discriminant and why is it important?
The discriminant determines the number and type of solutions a quadratic equation has; if it is positive, there are two real roots (two x-intercepts).
What happens if the discriminant equals zero?
The quadratic has exactly one real root, so the graph touches the x-axis at one point (a repeated or 'double' root).
What happens if the discriminant is negative?
There are no real roots, so the graph does not cross the x-axis.
Does changing m only affect the y-intercept?
Changing m shifts the whole parabola up or down, which also affects the number of x-intercepts.
How can I quickly determine the number of x-intercepts for a quadratic graph?
Calculate the discriminant (b^2 - 4ac); positive values mean two x-intercepts, zero means one, and negative means none.
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