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