solve by completing the square
324x^2+162x+1=0
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
To solve the quadratic equation by completing the square, follow the steps below:
Step 1: Move the constant term to the other side of the equation:
324x^2 + 162x = -1
Step 2: Divide the entire equation by the coefficient of x^2 (324):
x^2 + (162/324)x = -1/324
Step 3: Move the coefficient of x (162/324) to half of its value, squared, to the other side of the equation:
x^2 + (162/324)x + (162/324)^2 = -1/324 + (162/324)^2
Step 4: Simplify the right side of the equation:
x^2 + (162/324)x + (81/324)^2 = -1/324 + (81^2/324^2)
x^2 + (162/324)x + (81/324)^2 = -1/324 + 6561/104976
Step 5: Simplify further:
x^2 + (162/324)x + (81/324)^2 = -10651/104976
Step 6: Rewrite the left side of the equation as a perfect square:
(x + 81/324)^2 = -10651/104976
Step 7: Take the square root of both sides to solve for x:
(x + 81/324) = ± sqrt(-10651/104976)
Step 8: Simplify the right side of the equation:
(x + 81/324) = ± (i * sqrt(10651))/324
Step 9: Solve for x by subtracting 81/324 from both sides:
x = -81/324 ± (i * sqrt(10651))/324
Therefore, the solutions to the quadratic equation 324x^2 + 162x + 1 = 0, by completing the square, are:
x = (-81 ± i * sqrt(10651))/324
To solve the quadratic equation 324x^2 + 162x + 1 = 0 by completing the square, follow these steps:
Step 1: Make sure the coefficient of x^2 is 1. In this case, we already have a coefficient of 1, so we can proceed.
Step 2: Move the constant term (the number without x) to the other side of the equation. In this case, move 1 to the right side:
324x^2 + 162x = -1
Step 3: Divide the entire equation by the coefficient of x^2. In this case, divide both sides by 324:
x^2 + (162/324)x = -1/324
Step 4: Take half of the coefficient of x and then square it. In this case, half of (162/324) is (1/2) and (1/2)^2 is 1/4:
x^2 + (162/324)x + 1/4 = -1/324 + 1/4
Step 5: Simplify the right side of the equation. On the right side, -1/324 + 1/4 = -1/324 + 81/324 = 80/324 = 10/41. Therefore, the equation becomes:
x^2 + (162/324)x + 1/4 = 10/41
Step 6: Rewrite the left side of the equation as a perfect square. Take the square root of the constant term (which is 1/4) and add it to both sides. In this case, the square root of 1/4 is 1/2:
(x + 1/2)^2 = 10/41 + 1/4
Step 7: Combine the right side of the equation. On the right side, the common denominator of 41 and 4 is 164, so:
(x + 1/2)^2 = (10/41)*4/4 + 1/4 = 40/164 + 41/164 = 81/164
Step 8: Take the square root of both sides of the equation:
√((x + 1/2)^2) = ±√(81/164)
Step 9: Simplify the right side of the equation. The square root of 81 is 9, and the square root of 164 cannot be simplified further:
x + 1/2 = ±(9/√164)
Step 10: Simplify the right side further. Multiply both the numerator and denominator of 9/√164 by √164 to get:
x + 1/2 = ±9√164/164
Step 11: Subtract 1/2 from both sides of the equation:
x = -1/2 ± 9√164/164
Therefore, the solutions to the quadratic equation 324x^2 + 162x + 1 = 0 are:
x = -1/2 + 9√164/164
x = -1/2 - 9√164/164