To solve the equation 324x^2 + 162x + 1 = 0 by completing the square, we follow these steps:
1. Divide the entire equation by the coefficient of x^2 to make the coefficient of x^2 1.
(324x^2 + 162x + 1)/324 = 0
Simplifying, we get: x^2 + (1/2)x + (1/324) = 0
2. Move the constant term, (1/324), to the other side of the equation.
x^2 + (1/2)x = -1/324
3. Take half of the coefficient of x, which is (1/2), square it, and add it to both sides of the equation. This step completes the square on the left side of the equation.
x^2 + (1/2)x + (1/4)^2 = -1/324 + (1/4)^2
Simplifying, we get: x^2 + (1/2)x + 1/16 = -1/324 + 1/16
4. Simplify the right side of the equation.
-1/324 + 1/16 = -1/324 + 81/324
Simplifying further, we get: -1/324 + 81/324 = 80/324
Therefore, the equation becomes: x^2 + (1/2)x + 1/16 = 80/324
5. Factor the left side of the equation if possible.
The left side of the equation, x^2 + (1/2)x + 1/16, is a perfect square trinomial and can be factored as (x + 1/4)^2.
6. The equation can be rewritten as:
(x + 1/4)^2 = 80/324
7. Take the square root of both sides of the equation to solve for x.
x + 1/4 = ±√(80/324)
8. Simplify the right side of the equation.
√(80/324) can be simplified as √(4/9) * √(20/9) = (2/3) * (2√5 / 3) = (4√5) / 9
9. Solve for x by subtracting 1/4 from both sides of the equation.
x = -1/4 ± (4√5) / 9
Therefore, the solutions to the equation 324x^2 + 162x + 1 = 0 are:
x = (-1 ± 4√5) / 9
solve by completing the square
324x^2+162x+1=0
1 answer
1 answer