To find the solutions to the quadratic equation 2x^2 + 5x + 3 = 0, we can use various methods such as factoring, completing the square, or using the quadratic formula. Let's explore each of these methods in detail:
1. Factoring:
In order to factor the quadratic equation, we need to find two numbers that multiply to give us the constant term (3) and add up to give us the coefficient of the linear term (5). In this case, the numbers are 3 and 1. Therefore, we can rewrite the equation as follows:
2x^2 + 3x + 2x + 3 = 0
Now, we can group the terms and factor by grouping:
(x(2x + 3) + 1(2x + 3)) = 0
Now, we can see that we have a common factor of (2x + 3):
(2x + 3)(x + 1) = 0
Setting each factor equal to zero gives us two possible solutions:
2x + 3 = 0 or x + 1 = 0
Solving these equations gives us:
2x = -3 or x = -1
Dividing both sides of the first equation by 2 gives us:
x = -3/2
Therefore, the solutions to the equation are x = -3/2 and x = -1.