To find the x-intercepts of the graph of the function [tex]\( f(x) = 2x^2 - 5x + 3 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] where the function equal zero, i.e., [tex]\( f(x) = 0 \)[/tex].
Here is the step-by-step process:
1. Set the function equal to zero:
[tex]\[
2x^2 - 5x + 3 = 0
\][/tex]
2. Solve the quadratic equation. A standard method to solve a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\( 2x^2 - 5x + 3 = 0 \)[/tex], the coefficients are [tex]\( a = 2 \)[/tex], [tex]\( b = -5 \)[/tex], and [tex]\( c = 3 \)[/tex].
3. Plug these coefficients into the quadratic formula:
[tex]\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}
\][/tex]
4. Simplify inside the square root:
[tex]\[
x = \frac{5 \pm \sqrt{25 - 24}}{4}
\][/tex]
[tex]\[
x = \frac{5 \pm \sqrt{1}}{4}
\][/tex]
5. Simplify the square root:
[tex]\[
x = \frac{5 \pm 1}{4}
\][/tex]
6. Split into two solutions:
[tex]\[
x = \frac{5 + 1}{4} = \frac{6}{4} = 1.5
\][/tex]
[tex]\[
x = \frac{5 - 1}{4} = \frac{4}{4} = 1
\][/tex]
So, the x-intercepts of the graph of [tex]\( f(x) = 2x^2 - 5x + 3 \)[/tex] are:
[tex]\[
x = 1 \quad \text{and} \quad x = 1.5
\][/tex]
These are the points where the graph crosses the x-axis. Thus, the x-intercepts are [tex]\( \boxed{1 \ and \ 1.5} \)[/tex].
