Solve the following quadratic equation by completing the square. Simplify the solutions and rationalize denominators, if necessary.

x2 − 9x + 14 = 0

To solve the quadratic equation x^2 - 9x + 14 = 0 by completing the square, we can follow the steps below:

Step 1: Move the constant term to the other side of the equation:
x^2 - 9x = -14

Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation. (In this case, the coefficient of x is -9.)
x^2 - 9x + (-9/2)^2 = -14 + (-9/2)^2
x^2 - 9x + 81/4 = -14 + 81/4

Step 3: Simplify the right side of the equation:
x^2 - 9x + 81/4 = -56/4 + 81/4
x^2 - 9x + 81/4 = 25/4

Step 4: Factor the perfect square trinomial on the left side:
(x - 9/2)^2 = 25/4

Step 5: Take the square root of both sides of the equation, remembering to consider both positive and negative roots:
x - 9/2 = ±√(25/4)

Step 6: Simplify the square root expression:
x - 9/2 = ±5/2

Step 7: Solve for x by adding 9/2 to both sides of the equation:
x = 9/2 ± 5/2

Step 8: Simplify the solutions by combining like terms:
x = (9 ± 5)/2

Step 9: Separate the solutions:
x = (9 + 5)/2 or x = (9 - 5)/2

Step 10: Simplify the solutions further:
x = 14/2 or x = 4/2

Step 11: Simplify the solutions even more:
x = 7 or x = 2

The solutions to the quadratic equation x^2 - 9x + 14 = 0, by completing the square, are x = 7 and x = 2.

To solve the quadratic equation x^2 - 9x + 14 = 0 by completing the square, follow these steps:

Step 1: Move the constant term to the other side of the equation.

x^2 - 9x = -14

Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation. This completes the square on the left side of the equation.

x^2 - 9x + (-9/2)^2 = -14 + (-9/2)^2

x^2 - 9x + 81/4 = -14 + 81/4

Step 3: Simplify the right side of the equation.

x^2 - 9x + 81/4 = -56/4 + 81/4

x^2 - 9x + 81/4 = 25/4

Step 4: Rewrite the left side of the equation as a binomial squared.

(x - 9/2)^2 = 25/4

Step 5: Take the square root of both sides of the equation.

√[(x - 9/2)^2] = ± √(25/4)

x - 9/2 = ± 5/2

Step 6: Solve for x by adding 9/2 to both sides.

x = 9/2 ± 5/2

Simplifying the solutions:

x = (9 + 5)/2 or x = (9 - 5)/2

x = 14/2 = 7 or x = 4/2 = 2

Rationalizing denominators (if necessary):

All the denominators are already rational, so no further rationalization is needed.

Therefore, the solutions to the quadratic equation x^2 - 9x + 14 = 0, after completing the square, simplifying, and rationalizing denominators if necessary, are x = 7 and x = 2.

x^2-9x+14 = 0

x^2 - 9x + (9/2)^2 = -14 + (9/2)^2
(x - 9/2)^2 = 25/4
x - 9/2 = ±5/2
x = 9/2 ± 5/2
x = 2 or 7