%5E1%2F2%3D3.jpeg "(2a^2+5a+2)^1/2=3")
Solving the Equation: √(2a² + 5a + 2) = 3
This article will guide you through the process of solving the equation √(2a² + 5a + 2) = 3.
1. Squaring Both Sides
To eliminate the square root, we square both sides of the equation:
[√(2a² + 5a + 2)]² = 3²
This simplifies to:
2a² + 5a + 2 = 9
2. Rearranging the Equation
Now, we need to rearrange the equation to set it equal to zero:
2a² + 5a + 2 - 9 = 0
2a² + 5a - 7 = 0
3. Factoring the Quadratic Equation
The equation is now a quadratic equation. We can factor it as follows:
(2a - 1)(a + 7) = 0
4. Solving for 'a'
For the product of two terms to be zero, at least one of them must be zero. Therefore:
- 2a - 1 = 0
- a + 7 = 0
Solving these equations, we get:
- a = 1/2
- a = -7
5. Checking for Extraneous Solutions
It's important to check our solutions by substituting them back into the original equation to ensure they are valid:
- For a = 1/2:
- √(2(1/2)² + 5(1/2) + 2) = √(1/2 + 5/2 + 2) = √(9/2) = 3/√2 ≠ 3
- For a = -7:
- √(2(-7)² + 5(-7) + 2) = √(98 - 35 + 2) = √(65) ≠ 3
Therefore, a = 1/2 is an extraneous solution and a = -7 is the only valid solution to the equation.
Therefore, the solution to the equation √(2a² + 5a + 2) = 3 is a = -7.