Answer :
To find the values of [tex]\( m \)[/tex] for which the graph of the quadratic equation [tex]\( y = 3x^2 + 7x + m \)[/tex] has two [tex]\( x \)[/tex]-intercepts, we need to consider the discriminant of the quadratic equation.
The discriminant for a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
For the quadratic equation to have two distinct [tex]\( x \)[/tex]-intercepts, the discriminant must be greater than 0.
Here, we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
Now, let's calculate the discriminant:
[tex]\[
D = 7^2 - 4 \times 3 \times m
\][/tex]
This simplifies to:
[tex]\[
D = 49 - 12m
\][/tex]
To ensure that the graph has two [tex]\( x \)[/tex]-intercepts, we need:
[tex]\[
49 - 12m > 0
\][/tex]
Solving this inequality for [tex]\( m \)[/tex]:
1. Subtract 49 from both sides:
[tex]\[
-12m > -49
\][/tex]
2. Divide each side by [tex]\(-12\)[/tex] (note that dividing by a negative number flips the inequality sign):
[tex]\[
m < \frac{49}{12}
\][/tex]
Calculating [tex]\(\frac{49}{12}\)[/tex] gives approximately 4.0833.
Therefore, for the quadratic equation to have two [tex]\( x \)[/tex]-intercepts, [tex]\( m \)[/tex] must be less than [tex]\(\frac{49}{12}\)[/tex]. Hence, the correct choice is:
[tex]\( m < \frac{49}{12} \)[/tex]
The discriminant for a quadratic equation of the form [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[
D = b^2 - 4ac
\][/tex]
For the quadratic equation to have two distinct [tex]\( x \)[/tex]-intercepts, the discriminant must be greater than 0.
Here, we have:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
Now, let's calculate the discriminant:
[tex]\[
D = 7^2 - 4 \times 3 \times m
\][/tex]
This simplifies to:
[tex]\[
D = 49 - 12m
\][/tex]
To ensure that the graph has two [tex]\( x \)[/tex]-intercepts, we need:
[tex]\[
49 - 12m > 0
\][/tex]
Solving this inequality for [tex]\( m \)[/tex]:
1. Subtract 49 from both sides:
[tex]\[
-12m > -49
\][/tex]
2. Divide each side by [tex]\(-12\)[/tex] (note that dividing by a negative number flips the inequality sign):
[tex]\[
m < \frac{49}{12}
\][/tex]
Calculating [tex]\(\frac{49}{12}\)[/tex] gives approximately 4.0833.
Therefore, for the quadratic equation to have two [tex]\( x \)[/tex]-intercepts, [tex]\( m \)[/tex] must be less than [tex]\(\frac{49}{12}\)[/tex]. Hence, the correct choice is:
[tex]\( m < \frac{49}{12} \)[/tex]
