Solving the Quadratic Equation: 2x² + (5/2)x  √3 = 0
This article will guide you through the steps to solve the quadratic equation 2x² + (5/2)x  √3 = 0. We'll utilize the quadratic formula to find the solutions.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool for finding the roots (or solutions) of any quadratic equation in the form ax² + bx + c = 0. The formula is:
x = (b ± √(b²  4ac)) / 2a
Where:
 a, b, and c are the coefficients of the quadratic equation.
Applying the Formula to Our Equation
Let's identify the coefficients in our equation:
 a = 2
 b = 5/2
 c = √3
Now, we can plug these values into the quadratic formula:
x = ((5/2) ± √((5/2)²  4 * 2 * √3)) / (2 * 2)
Simplifying the Expression
Let's break down the calculation stepbystep:

Simplify inside the square root: (5/2)²  4 * 2 * √3 = 25/4 + 8√3

Calculate the denominator: 2 * 2 = 4

Substitute the values back into the formula: x = (5/2 ± √(25/4 + 8√3)) / 4
Finding the Solutions
We have two possible solutions, one with a plus sign and one with a minus sign:
 x1 = (5/2 + √(25/4 + 8√3)) / 4
 x2 = (5/2  √(25/4 + 8√3)) / 4
These solutions represent the points where the graph of the quadratic equation intersects the xaxis.
Conclusion
By utilizing the quadratic formula, we have successfully found the two solutions for the equation 2x² + (5/2)x  √3 = 0. These solutions, although complex in appearance, represent the points where the equation intersects the xaxis.