Solve by completing the square:
x^2+9x+6=0
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term (6) to the right side of the equation by subtracting it from both sides:
x^2 + 9x = -6
Step 2: Take half of the coefficient of x (which is 9), square it, and add it to both sides of the equation. This completes the square on the left side:
x^2 + 9x + (9/2)^2 = -6 + (9/2)^2
x^2 + 9x + 81/4 = -6 + 81/4
Step 3: Simplify both sides of the equation:
x^2 + 9x + 81/4 = -24/4 + 81/4
x^2 + 9x + 81/4 = 57/4
Step 4: Factor the left side of the equation:
(x + 9/2)^2 = 57/4
Step 5: Take the square root of both sides:
√((x + 9/2)^2) = ±√(57/4)
x + 9/2 = ±√(57/4)
Step 6: Solve for x by subtracting 9/2 from both sides:
x = -9/2 ±√(57/4)
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Set up the equation in the form "ax^2 + bx + c = 0".
In this case, the equation is already in that form: x^2 + 9x + 6 = 0.
Step 2: Identify the coefficient of x^2, the term with the x coefficient, and the constant term.
In this equation, a = 1 (coefficient of x^2), b = 9 (coefficient of x), and c = 6 (constant term).
Step 3: Divide the entire equation by the coefficient of x^2 (a) if it's not equal to 1.
Since a = 1, no division is necessary.
Step 4: Move the constant term (c) to the right side of the equation.
The equation becomes: x^2 + 9x = -6.
Step 5: Take half of the coefficient of x (b/2), square it [(b/2)^2], and add it to both sides of the equation.
In this case, (9/2)^2 = 81/4. Adding 81/4 to both sides of the equation:
x^2 + 9x + 81/4 = -6 + 81/4.
Step 6: Simplify the right side of the equation.
-6 + 81/4 = -24/4 + 81/4 = 57/4.
Step 7: Simplify the left side of the equation as a perfect square trinomial.
(x + b/2)^2 = (x + 9/2)^2 = x^2 + 9x + (9/2)^2 = x^2 + 9x + 81/4.
Step 8: Set the equation as: (x + b/2)^2 = right side of the equation.
(x + 9/2)^2 = 57/4.
Step 9: Take the square root of both sides of the equation.
x + 9/2 = ±√(57/4).
Step 10: Solve for x by isolating it on one side of the equation.
x = -9/2 ± √(57/4).
So the solutions to the quadratic equation x^2 + 9x + 6 = 0, obtained by completing the square, are:
x = -9/2 + √(57/4)
x = -9/2 - √(57/4)