To determine how many times the quadratic function [tex]\( y = x^2 + 10x + 25 \)[/tex] intersects the [tex]\( x \)[/tex]-axis, we need to find the roots of the quadratic equation. The roots can be found by solving the equation [tex]\( x^2 + 10x + 25 = 0 \)[/tex].
A quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] intersects the [tex]\( x \)[/tex]-axis at points where [tex]\( y = 0 \)[/tex]. To solve this, we use the quadratic formula:
[tex]\[
x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}
\][/tex]
In the given equation [tex]\( y = x^2 + 10x + 25 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\( c = 25 \)[/tex]
The next step is to calculate the discriminant ([tex]\( \Delta \)[/tex]) of the quadratic equation, which is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values [tex]\( a = 1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = 25 \)[/tex]:
[tex]\[
\Delta = 10^2 - 4(1)(25) = 100 - 100 = 0
\][/tex]
The value of the discriminant ([tex]\( \Delta \)[/tex]) tells us the nature of the roots:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real roots.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real root (a repeated root).
- If [tex]\( \Delta < 0 \)[/tex], there are no real roots (the roots are complex).
In this case, since [tex]\( \Delta = 0 \)[/tex], the quadratic equation [tex]\( x^2 + 10x + 25 = 0 \)[/tex] has exactly one real root. This indicates that the graph of the quadratic function touches the [tex]\( x \)[/tex]-axis at exactly one point.
Therefore, the quadratic function [tex]\( y = x^2 + 10x + 25 \)[/tex] intersects the [tex]\( x \)[/tex]-axis:
B. 1
