To determine the [tex]\( x \)[/tex]-intercepts of the quadratic function [tex]\( y = 9x^2 - 30x + 25 \)[/tex], we need to find the values of [tex]\( x \)[/tex] for which [tex]\( y \)[/tex] is zero. In other words, we need to solve the equation:
[tex]\[ 9x^2 - 30x + 25 = 0 \][/tex]
We can find the [tex]\( x \)[/tex]-intercepts using the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our specific quadratic equation [tex]\( 9x^2 - 30x + 25 \)[/tex], we have:
- [tex]\( a = 9 \)[/tex]
- [tex]\( b = -30 \)[/tex]
- [tex]\( c = 25 \)[/tex]
First, we calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-30)^2 - 4 \cdot 9 \cdot 25 \][/tex]
[tex]\[ \Delta = 900 - 900 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
Since the discriminant [tex]\(\Delta\)[/tex] is zero, there is exactly one real solution to this quadratic equation, meaning that the equation has one repeated root (the parabola touches the x-axis at exactly one point).
Next, we use the quadratic formula to find the solution:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{30 \pm \sqrt{0}}{2 \cdot 9} \][/tex]
[tex]\[ x = \frac{30 \pm 0}{18} \][/tex]
[tex]\[ x = \frac{30}{18} \][/tex]
[tex]\[ x = \frac{5}{3} \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts of the quadratic function [tex]\( y = 9x^2 - 30x + 25 \)[/tex] are both located at:
[tex]\[ x = \frac{5}{3} \][/tex]
Thus, the correct answer from the given options is:
[tex]\[ x = \frac{5}{3} \text{ and } x = \frac{5}{3} \][/tex]
