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