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]$m>\frac{25}{3}$[/tex]
B. [tex]$m<\frac{25}{3}$[/tex]
C. [tex]$m<\frac{49}{12}$[/tex]
D. [tex]$m>\frac{49}{12}$[/tex]
Answer (2)
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 of the quadratic equation is calculated as b 2 − 4 a c = 7 2 − 4 ( 3 ) ( m ) = 49 − 12 m .
For two distinct real roots, the discriminant must be greater than 0, so 0"> 49 − 12 m > 0 .
Solving the inequality for m gives m < 12 49 . Therefore, the final answer is m < 12 49 .
Explanation
Problem Analysis 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.
Applying 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 this case, a = 3 , b = 7 , and c = m . So the discriminant is 7 2 − 4 ( 3 ) ( m ) = 49 − 12 m .
Setting up the Inequality We want the discriminant 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 : 12m"> 49 > 12 m m"> 12 49 > m So, 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
Understanding quadratic equations and their intercepts is crucial in various real-world applications. For instance, when designing a parabolic reflector for a flashlight, knowing the conditions for the parabola to intersect the x-axis at two points helps determine the optimal placement of the light source to maximize the beam's focus. Similarly, in projectile motion, the x-intercepts of the parabolic trajectory represent the points where the projectile lands, and controlling these intercepts is essential for accurate targeting. The discriminant helps engineers and physicists predict these outcomes by analyzing the equation's coefficients.
The quadratic equation y = 3 x 2 + 7 x + m has two x -intercepts when m < 12 49 . Thus, the correct answer is option C. This means for any value of m less than 12 49 , the graph will intersect the x-axis at two points.
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