How can I factor 18x^2+x=5 by completing the square?
To factor the quadratic equation 18x^2 + x = 5 by completing the square, follow these steps:
Step 1: Move the constant term to the other side
18x^2 + x - 5 = 0
Step 2: Divide the entire equation by the coefficient of x^2 so the leading coefficient is 1.
x^2 + (1/18)x - 5/18 = 0
Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation.
The coefficient of x is (1/18), halving it gives (1/36), and squaring it gives (1/1296).
Add (1/1296) to both sides:
x^2 + (1/18)x + (1/1296) - (5/18) = (1/1296) - (5/18)
Step 4: Simplify the equation.
x^2 + (1/18)x + (1/1296) - (5/18) = -719/1296
Step 5: Factor the left side of the equation.
(x + (1/36))^2 = -719/1296
Step 6: Take the square root of both sides.
x + (1/36) = ±√(-719/1296)
Step 7: Simplify the square root.
x + (1/36) = ±(√719/36√2)
Step 8: Isolate x by subtracting (1/36) from both sides.
x = -1/36 ± (√719/36√2)
Now, the quadratic equation 18x^2 + x = 5 has been factored by completing the square, and the solutions for x have been obtained.