Solve the Quadratic Equation: (1/2)x² - 3x + 2.5 = 0

To solve the quadratic equation $\frac{1}{2}x^2 - 3x + 2\frac{1}{2} = 0$, we will follow these steps:

• Step 1: Convert the mixed number to an improper fraction or decimal for uniformity in calculations.
• Step 2: Identify the coefficients $a$, $b$, and $c$.
• Step 3: Apply the quadratic formula to find the roots $x_1$ and $x_2$.

Step 1: Rewriting the equation with decimals, we have:

$\frac{1}{2}x^2 - 3x + 2.5 = 0$

Step 2: The equation is already in standard form, where $a = \frac{1}{2}$, $b = -3$, and $c = 2.5$.

Step 3: Apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Calculate the discriminant ($\Delta$):

$\Delta = b^2 - 4ac = (-3)^2 - 4 \cdot \frac{1}{2} \cdot 2.5 = 9 - 5 = 4$

Since the discriminant is positive, there are two real solutions.

Calculate the roots:

$x = \frac{-(-3) \pm \sqrt{4}}{2 \cdot \frac{1}{2}}$

$x = \frac{3 \pm 2}{1}$

Thus, the solutions are:

• $x_1 = \frac{3 + 2}{1} = 5$
• $x_2 = \frac{3 - 2}{1} = 1$

Therefore, the solutions to the equation are $x_1 = 5$ and $x_2 = 1$.

From the provided choices, the correct answer is:

$x_1=5,x_2=1$