To determine how many times the quadratic function [tex]\( y = 2x^2 + 7x + 6 \)[/tex] intersects the [tex]\( x \)[/tex]-axis, we need to analyze its roots. The roots of a quadratic function are found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
1. First, identify the coefficients:
[tex]\[
a = 2, \quad b = 7, \quad c = 6
\][/tex]
2. Compute the discriminant ([tex]\( \Delta \)[/tex]), which is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
3. Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\Delta = 7^2 - 4 \cdot 2 \cdot 6
\][/tex]
[tex]\[
\Delta = 49 - 48
\][/tex]
[tex]\[
\Delta = 1
\][/tex]
4. Analyze the discriminant:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real roots.
- If [tex]\( \Delta = 0 \)[/tex], there is one real root.
- If [tex]\( \Delta < 0 \)[/tex], there are no real roots.
Given that [tex]\( \Delta = 1 \)[/tex] (which is greater than 0), it means that the quadratic function has two distinct real roots. Therefore, the function intersects the [tex]\( x \)[/tex]-axis at two points.
Thus, the quadratic function [tex]\( y = 2x^2 + 7x + 6 \)[/tex] intersects the [tex]\( x \)[/tex]-axis 2 times.
The correct answer is:
C. 2
