To determine the number of [tex]\( x \)[/tex]-intercepts of the graph of the quadratic function [tex]\( y = 2x^2 - 4x + 2 \)[/tex], we should solve the equation [tex]\( 2x^2 - 4x + 2 = 0 \)[/tex].
We can solve this quadratic equation using various methods, such as factoring, completing the square, or the quadratic formula. For this instance, let's use the quadratic formula which is given by:
[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].
Given the equation [tex]\( 2x^2 - 4x + 2 = 0 \)[/tex], we can identify the coefficients:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = 2 \)[/tex]
Now we apply these values to the quadratic formula:
[tex]\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{16 - 16}}{4} \][/tex]
[tex]\[ x = \frac{4 \pm \sqrt{0}}{4} \][/tex]
[tex]\[ x = \frac{4 \pm 0}{4} \][/tex]
[tex]\[ x = \frac{4}{4} \][/tex]
[tex]\[ x = 1 \][/tex]
Since the discriminant ([tex]\( b^2 - 4ac \)[/tex]) is zero, this means the quadratic equation has exactly one real solution. Therefore, the graph of the quadratic function intersects the [tex]\( x \)[/tex]-axis at exactly one point.
Thus, the number of [tex]\( x \)[/tex]-intercepts of the graph of [tex]\( y = 2x^2 - 4x + 2 \)[/tex] is:
C. 1
