Questions: Without graphing, find the x - and y-intercepts for f(x). f(x) = x^2 - 5x + 3 Select the correct choice below and fill in any answer boxes in your choice. A. There is no x-intercept. B. The x-intercept(s) are (Type an integer or decimal rounded to the nearest tenth as needed. Use a comma to separate answers as needed.)

Solution Steps

To find the \(x\)-intercepts of the quadratic function \(f(x) = x^2 - 5x + 3\), we need to solve the equation \(f(x) = 0\). This involves finding the roots of the quadratic equation \(x^2 - 5x + 3 = 0\). We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. For the \(y\)-intercept, we evaluate \(f(x)\) at \(x = 0\).

For the quadratic function \( f(x) = x^2 - 5x + 3 \), we first calculate the discriminant using the formula: \[ D = b^2 - 4ac \] Substituting the values \( a = 1 \), \( b = -5 \), and \( c = 3 \): \[ D = (-5)^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13 \]

Since the discriminant \( D = 13 \) is positive, there are two real \( x \)-intercepts. We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{-(-5) \pm \sqrt{13}}{2 \cdot 1} = \frac{5 \pm \sqrt{13}}{2} \] Thus, the \( x \)-intercepts are: \[ x_1 = \frac{5 + \sqrt{13}}{2}, \quad x_2 = \frac{5 - \sqrt{13}}{2} \]

The \( y \)-intercept is found by evaluating \( f(0) \): \[ f(0) = 0^2 - 5 \cdot 0 + 3 = 3 \] Thus, the \( y \)-intercept is \( 3 \).

Final Answer