-2x%5E(2)%2B(5)%2F(2)x-sqrt(3)%3D0.jpeg "(ii) 2x^(2)+(5)/(2)x-sqrt(3)=0")
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 step-by-step:
-
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 x-axis.
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 x-axis.