Solve the equation by completing the square.
2x^2 + 9x + 1 = 0
x^2+9/2 x + 1/2 =0
x^2+9/2 x + 81/16 = 81/16 - 8/16
(x+9/4)^2= (1/4 sqrt 73)^2
x=+-1/4 sqrt73 -9/4
2x^2 +9x = -1
x^2 + 9x/2 +81/16 = -1/2 +81/16
(x +9/4)^2 = 73/16
x + 9/4 = -+sqrt(73/16)
x = (-9 - sqrt(73))/4
x = (-9 + sqrt(73))/4
To solve the equation by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation to create a space for completing the square:
2x^2 + 9x = -1
Step 2: Divide the coefficient of x^2 by 2, then square it. Add this value on both sides of the equation:
2x^2 + 9x + (9/2)^2 = -1 + (9/2)^2
Simplifying:
2x^2 + 9x + 81/4 = -1 + 81/4
2x^2 + 9x + 81/4 = -4/4 + 81/4
2x^2 + 9x + 81/4 = 77/4
Step 3: Express the left side as a perfect square trinomial.
To do this, we need to find the perfect square trinomial that can be factored as (ax + b)^2.
We take the coefficient of x (which is 9), divide it by 2, and square it:
(9/2)^2 = 81/4
So, our perfect square trinomial is:
(2x + 9/2)^2 = 77/4
Step 4: Take the square root of both sides to solve for x:
2x + 9/2 = ±√(77/4)
Step 5: Solve for x:
2x + 9/2 = ±(√77)/2
For the positive square root:
2x + 9/2 = (√77)/2
2x = -9/2 + (√77)/2
2x = (-9 + √77)/2
x = (-9 + √77)/4
For the negative square root:
2x + 9/2 = - (√77)/2
2x = -9/2 - (√77)/2
2x = (-9 - √77)/2
x = (-9 - √77)/4
Therefore, the solutions to the equation are:
x = (-9 + √77)/4 and x = (-9 - √77)/4