For what values of [tex]$m$[/tex] does the graph of [tex]$y=3 x^2+7 x+m$[/tex] have two [tex]$x$[/tex]-intercepts?
A. [tex]$m\ \textgreater \ \frac{25}{3}$[/tex]
B. [tex]$m\ \textless \ \frac{25}{3}$[/tex]
C. [tex]$m\ \textless \ \frac{49}{12}$[/tex]
D. [tex]$m\ \textgreater \ \frac{49}{12}$[/tex]

The values of m for which the graph of y = 3 x 2 + 7 x + m has two x-intercepts must satisfy m < 12 49 ​ . Therefore, the correct answer is option C.
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Answered by Anonymous | 2025-07-04

The problem requires finding the values of m for which the quadratic equation 3 x 2 + 7 x + m = 0 has two distinct real roots.
The discriminant b 2 − 4 a c must be greater than 0 for two distinct real roots.
Calculate the discriminant: 7 2 − 4 ( 3 ) ( m ) = 49 − 12 m .
Solve the inequality 0"> 49 − 12 m > 0 to find m < 12 49 ​ .
The final answer is m < 12 49 ​ ​ .

Explanation

Understanding the Problem We are given the quadratic equation y = 3 x 2 + 7 x + m and we want to find the values of m for which the graph has two x -intercepts. This means we want to find the values of m for which the equation 3 x 2 + 7 x + m = 0 has two distinct real roots.

Using the Discriminant A quadratic equation a x 2 + b x + c = 0 has two distinct real roots if and only if its discriminant, b 2 − 4 a c , is greater than 0. In our equation, a = 3 , b = 7 , and c = m .

Setting up the Inequality The discriminant is 7 2 − 4 ( 3 ) ( m ) = 49 − 12 m . We want this to be greater than 0, so we have the inequality 0"> 49 − 12 m > 0 .

Solving for m Now, we solve the inequality for m :


0"> 49 − 12 m > 0
12m"> 49 > 12 m
m < 12 49 ​

Final Answer Therefore, the graph of y = 3 x 2 + 7 x + m has two x -intercepts when m < 12 49 ​ .

Examples
Imagine you are designing a parabolic bridge. The equation y = 3 x 2 + 7 x + m describes the shape of the bridge, where y is the height and x is the horizontal distance. The x -intercepts represent where the bridge touches the ground. To ensure the bridge touches the ground at two distinct points, you need to choose a value for m (a design parameter) such that m < 12 49 ​ . This ensures the bridge has the desired shape and stability.

Answered by GinnyAnswer | 2025-07-04