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 formula. The discriminant is given by [tex]\( b^2 - 4ac \)[/tex].
In a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the discriminant determines the nature of the roots (or [tex]\( x \)[/tex]-intercepts):
- If the discriminant is greater than 0, the quadratic has two distinct [tex]\( x \)[/tex]-intercepts.
- If the discriminant is equal to 0, it has exactly one [tex]\( x \)[/tex]-intercept (a repeated root).
- If the discriminant is less than 0, it has no real [tex]\( x \)[/tex]-intercepts.
For the quadratic equation [tex]\( y = 3x^2 + 7x + m \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
The discriminant is:
[tex]\[
b^2 - 4ac = 7^2 - 4 \times 3 \times m = 49 - 12m
\][/tex]
We need this discriminant to be greater than 0 for the quadratic to have two distinct [tex]\( x \)[/tex]-intercepts:
[tex]\[
49 - 12m > 0
\][/tex]
Solving this inequality:
1. Subtract 49 from both sides:
[tex]\[
-12m > -49
\][/tex]
2. Divide by -12. Remember, when we divide by a negative number, the inequality sign flips:
[tex]\[
m < \frac{49}{12}
\][/tex]
So, the condition for the quadratic to have two [tex]\( x \)[/tex]-intercepts is [tex]\( m < \frac{49}{12} \)[/tex].
Therefore, the correct answer is:
- [tex]\( m < \frac{49}{12} \)[/tex]
In a quadratic equation [tex]\( ax^2 + bx + c \)[/tex], the discriminant determines the nature of the roots (or [tex]\( x \)[/tex]-intercepts):
- If the discriminant is greater than 0, the quadratic has two distinct [tex]\( x \)[/tex]-intercepts.
- If the discriminant is equal to 0, it has exactly one [tex]\( x \)[/tex]-intercept (a repeated root).
- If the discriminant is less than 0, it has no real [tex]\( x \)[/tex]-intercepts.
For the quadratic equation [tex]\( y = 3x^2 + 7x + m \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
The discriminant is:
[tex]\[
b^2 - 4ac = 7^2 - 4 \times 3 \times m = 49 - 12m
\][/tex]
We need this discriminant to be greater than 0 for the quadratic to have two distinct [tex]\( x \)[/tex]-intercepts:
[tex]\[
49 - 12m > 0
\][/tex]
Solving this inequality:
1. Subtract 49 from both sides:
[tex]\[
-12m > -49
\][/tex]
2. Divide by -12. Remember, when we divide by a negative number, the inequality sign flips:
[tex]\[
m < \frac{49}{12}
\][/tex]
So, the condition for the quadratic to have two [tex]\( x \)[/tex]-intercepts is [tex]\( m < \frac{49}{12} \)[/tex].
Therefore, the correct answer is:
- [tex]\( m < \frac{49}{12} \)[/tex]
