Consider the general quadratic equation ax^2 + bx + c = 0, where a is not equal to 0. Solve for x by completing the square.

ax^2 + bx + c = 0

a(x^2 + b/a x) = -c
a(x^2 + b/a x + (b/2a)^2) = -c + a(b/2a)^2
a(x + b/2a)^2 = b^2/4a - c
(x + b/2a)^2 = (b^2-4ac)/4a^2
x + b/2a = ±√(b^2-4ac)/2a
x = (-b±√(b^2-4ac))/2a

Thank you so very much

To solve the quadratic equation by completing the square, follow these steps:

Step 1: Make sure the quadratic equation is in standard form, ax^2 + bx + c = 0, with a, b, and c being constants. If necessary, rearrange the equation to put it in this form.

Step 2: Divide the entire equation by the coefficient of x^2 (a) to simplify it. This will give us the equation: x^2 + (b/a)x + (c/a) = 0.

Step 3: In the equation, the coefficient of x (b/a) should be split in two, taking half of it. Rewrite the equation as follows: x^2 + (b/a)x + (b/2a)^2 - (b/2a)^2 + (c/a) = 0.

Step 4: The first three terms, x^2 + (b/a)x + (b/2a)^2, forms a perfect square trinomial, which can be factored as (x + b/2a)^2. Rewrite the equation accordingly: (x + b/2a)^2 - (b/2a)^2 + (c/a) = 0.

Step 5: Simplify the expression (b/2a)^2 - (c/a) to ((b/2a)^2 - (c/a))(a/a) and get a common denominator: (b^2 - 4ac) / (4a^2).

Step 6: Substitute the simplified expression back into the equation: (x + b/2a)^2 - (b^2 - 4ac) / (4a^2) = 0.

Step 7: Add (b^2 - 4ac) / (4a^2) to both sides of the equation to isolate the squared term: (x + b/2a)^2 = (b^2 - 4ac) / (4a^2).

Step 8: Take the square root of both sides of the equation to obtain: x + b/2a = ± √((b^2 - 4ac) / (4a^2)).

Step 9: Subtract b/2a from both sides to solve for x: x = (-b ± √(b^2 - 4ac)) / (2a).

Thus, the solutions to the quadratic equation ax^2 + bx + c = 0 obtained by completing the square are: x = (-b + √(b^2 - 4ac)) / (2a) and x = (-b - √(b^2 - 4ac)) / (2a).

To solve the quadratic equation ax^2 + bx + c = 0 by completing the square, follow these steps:

Step 1: Ensure that the coefficient of x^2 (a) is not equal to zero. If a = 0, the equation is not quadratic.

Step 2: Divide both sides of the equation by the coefficient of x^2 (a). This will result in a simplified form of the equation: x^2 + (b/a)x + c/a = 0.

Step 3: Move the constant term (c/a) to the right side of the equation: x^2 + (b/a)x = -c/a.

Step 4: Add the square of half of the coefficient of x (b/2a)^2 to both sides of the equation. This will complete the square on the left side: x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2.

Step 5: Simplify the right side of the equation: x^2 + (b/a)x + (b/2a)^2 = (-4ac + b^2) / (4a^2).

Step 6: Rewrite the left side of the equation as a perfect square: (x + b/2a)^2 = (-4ac + b^2) / (4a^2).

Step 7: Take the square root of both sides of the equation: x + b/2a = ±√[(-4ac + b^2)/(4a^2)].

Step 8: Simplify the square root expression: x + b/2a = ±√[-4ac + b^2] / (2a).

Step 9: Solve for x by subtracting b/2a from both sides of the equation: x = (-b ± √[b^2 - 4ac]) / (2a).

Therefore, the solutions to the quadratic equation ax^2 + bx + c = 0 by completing the square are x = (-b ± √[b^2 - 4ac]) / (2a).