Use the quadratic formula to find the exact roots of 2x² = 5x + 2

2x^2 - 5x - 2 = 0

x = [5±√(25+16)]/(2*2)

To find the exact roots of a quadratic equation using the quadratic formula, we first need to ensure that the equation is in the standard form, which is ax² + bx + c = 0. In the given equation, 2x² = 5x + 2, we see that it is already in the standard form.

The quadratic formula states that for any quadratic equation in the form ax² + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b² - 4ac)) / (2a)

Let's apply the quadratic formula to solve the equation 2x² = 5x + 2.

In this case, a = 2, b = -5, and c = -2. Substituting these values into the quadratic formula, we get:
x = (-(5) ± √((-5)² - 4(2)(-2))) / (2(2))

Simplifying further, we have:
x = (-(-5) ± √(25 + 16)) / 4
= (5 ± √(41)) / 4

Therefore, the exact roots of the equation 2x² = 5x + 2 are:
x₁ = (5 + √41) / 4
x₂ = (5 - √41) / 4

These are the two solutions to the equation.