Answer :
To determine 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 general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. The discriminant [tex]\( \Delta \)[/tex] of the quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the quadratic equation to have two distinct real [tex]\( x \)[/tex]-intercepts, the discriminant must be greater than zero:
[tex]\[ \Delta > 0 \][/tex]
In our equation [tex]\( y = 3x^2 + 7x + m \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
Substitute these values into the discriminant formula:
1. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = 7^2 = 49 \][/tex]
2. Calculate [tex]\( 4ac \)[/tex]:
[tex]\[ 4ac = 4 \times 3 \times m = 12m \][/tex]
3. Set up the inequality for the discriminant:
[tex]\[ \Delta = 49 - 12m > 0 \][/tex]
4. Solve the inequality for [tex]\( m \)[/tex]:
- Subtract 49 from both sides:
[tex]\[ 49 > 12m \][/tex]
- Divide each side by 12:
[tex]\[ \frac{49}{12} > m \][/tex]
So, the values of [tex]\( m \)[/tex] that make the graph of the quadratic equation have two [tex]\( x \)[/tex]-intercepts are [tex]\( m < \frac{49}{12} \)[/tex]. Thus, the correct choice from the given options is:
[tex]\( m < \frac{49}{12} \)[/tex]
The general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. The discriminant [tex]\( \Delta \)[/tex] of the quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For the quadratic equation to have two distinct real [tex]\( x \)[/tex]-intercepts, the discriminant must be greater than zero:
[tex]\[ \Delta > 0 \][/tex]
In our equation [tex]\( y = 3x^2 + 7x + m \)[/tex]:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 7 \)[/tex]
- [tex]\( c = m \)[/tex]
Substitute these values into the discriminant formula:
1. Calculate [tex]\( b^2 \)[/tex]:
[tex]\[ b^2 = 7^2 = 49 \][/tex]
2. Calculate [tex]\( 4ac \)[/tex]:
[tex]\[ 4ac = 4 \times 3 \times m = 12m \][/tex]
3. Set up the inequality for the discriminant:
[tex]\[ \Delta = 49 - 12m > 0 \][/tex]
4. Solve the inequality for [tex]\( m \)[/tex]:
- Subtract 49 from both sides:
[tex]\[ 49 > 12m \][/tex]
- Divide each side by 12:
[tex]\[ \frac{49}{12} > m \][/tex]
So, the values of [tex]\( m \)[/tex] that make the graph of the quadratic equation have two [tex]\( x \)[/tex]-intercepts are [tex]\( m < \frac{49}{12} \)[/tex]. Thus, the correct choice from the given options is:
[tex]\( m < \frac{49}{12} \)[/tex]
