Answer :
To determine the values of \( m \) for which the graph of the quadratic equation \( y = 3x^2 + 7x + m \) has two \( x \)-intercepts, we need to examine the discriminant of the quadratic equation.
A quadratic equation in standard form \( ax^2 + bx + c = 0 \) has a discriminant \(\Delta\), given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
The nature of the roots of the quadratic equation depends on the value of the discriminant \(\Delta\):
- If \(\Delta > 0\), the equation has two distinct real roots, which implies the graph has two \( x \)-intercepts.
- If \(\Delta = 0\), the equation has exactly one real root, which implies the graph has one \( x \)-intercept (the vertex of the parabola touches the \( x \)-axis).
- If \(\Delta < 0\), the equation has no real roots, which implies the graph has no \( x \)-intercepts.
For the given equation \( y = 3x^2 + 7x + m \):
[tex]\[ a = 3, \quad b = 7, \quad c = m \][/tex]
We want the discriminant to be greater than zero (\(\Delta > 0\)) for the graph to have two \( x \)-intercepts:
[tex]\[ \Delta = b^2 - 4ac > 0 \][/tex]
Substituting the values of \( a \), \( b \), and \( c \):
[tex]\[ 7^2 - 4 \cdot 3 \cdot m > 0 \][/tex]
[tex]\[ 49 - 12m > 0 \][/tex]
Solving for \( m \):
[tex]\[ 49 > 12m \][/tex]
[tex]\[ \frac{49}{12} > m \][/tex]
[tex]\[ m < \frac{49}{12} \][/tex]
Therefore, the quadratic equation \( y = 3x^2 + 7x + m \) has two \( x \)-intercepts when \( m < \frac{49}{12} \).
So, the correct answer is:
[tex]\[ m < \frac{49}{12} \][/tex]
A quadratic equation in standard form \( ax^2 + bx + c = 0 \) has a discriminant \(\Delta\), given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
The nature of the roots of the quadratic equation depends on the value of the discriminant \(\Delta\):
- If \(\Delta > 0\), the equation has two distinct real roots, which implies the graph has two \( x \)-intercepts.
- If \(\Delta = 0\), the equation has exactly one real root, which implies the graph has one \( x \)-intercept (the vertex of the parabola touches the \( x \)-axis).
- If \(\Delta < 0\), the equation has no real roots, which implies the graph has no \( x \)-intercepts.
For the given equation \( y = 3x^2 + 7x + m \):
[tex]\[ a = 3, \quad b = 7, \quad c = m \][/tex]
We want the discriminant to be greater than zero (\(\Delta > 0\)) for the graph to have two \( x \)-intercepts:
[tex]\[ \Delta = b^2 - 4ac > 0 \][/tex]
Substituting the values of \( a \), \( b \), and \( c \):
[tex]\[ 7^2 - 4 \cdot 3 \cdot m > 0 \][/tex]
[tex]\[ 49 - 12m > 0 \][/tex]
Solving for \( m \):
[tex]\[ 49 > 12m \][/tex]
[tex]\[ \frac{49}{12} > m \][/tex]
[tex]\[ m < \frac{49}{12} \][/tex]
Therefore, the quadratic equation \( y = 3x^2 + 7x + m \) has two \( x \)-intercepts when \( m < \frac{49}{12} \).
So, the correct answer is:
[tex]\[ m < \frac{49}{12} \][/tex]
