The goal of this problem is to solve the equation by completing the square.
x^2-9x+14=0
Step 1: Move the constant term to the right side of the equation:
x^2 - 9x = -14
Step 2: Take half of the coefficient of x (-9) and square it:
(-9/2)^2 = 81/4
Step 3: Add the result from Step 2 (81/4) to both sides of the equation:
x^2 - 9x + 81/4 = -14 + 81/4
This can be simplified to:
(x - 9/2)^2 = -14 + 81/4
Step 4: Simplify the right side of the equation:
(x - 9/2)^2 = -56/4 + 81/4
Which becomes:
(x - 9/2)^2 = 25/4
Step 5: Take the square root of both sides of the equation:
√(x - 9/2)^2 = ±√(25/4)
Which simplifies to:
x - 9/2 = ±5/2
Step 6: Solve for x by adding 9/2 to both sides of the equation:
x = 9/2 ± 5/2
This can be further simplified to:
x = (9 ± 5)/2
So the solutions to the equation x^2 - 9x + 14 = 0 are:
x = (9 + 5)/2 = 7
x = (9 - 5)/2 = 2
To solve the equation x^2 - 9x + 14 = 0 by completing the square, you can follow these steps:
Step 1: Rewrite the equation in the form (x - h)^2 = k.
- To do this, move the constant term (14 in this case) to the right side of the equation.
x^2 - 9x = -14
Step 2: Take half of the coefficient of x and square it. Add that value to both sides of the equation.
- In this case, the coefficient of x is -9, so half of it is -9/2, and when squared, it becomes 81/4.
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 left side of the equation.
(x - 9/2)^2 = 25/4
Step 5: Take the square root of both sides of the equation.
√((x - 9/2)^2) = ±√(25/4)
Step 6: Solve for x by adding or subtracting the value on the right side.
x - 9/2 = ±5/2
Step 7: Solve for x.
x = 9/2 ± 5/2
Simplifying the equation gives two possible solutions:
x = 7/2 or x = 11/2
Therefore, the solutions to the equation x^2 - 9x + 14 = 0 are x = 7/2 and x = 11/2.