Complete the square to solve the following quadratic equation: x^2 - 22 + 1 = 15.
A. x = -11 + 3 square root of 15, -11 - 3 square root of 15
B. x = 11 + 3 square root of 15, 11 - 3 square root of 15
C. x = -11 + square root of 14, -11 - square root of 14
D. x = 11 + square root of 14, 11 - square root of 14
To solve the quadratic equation x^2 - 22x + 1 = 15, we first move all terms to one side of the equation:
x^2 - 22x + 1 - 15 = 0.
Next, we simplify the left side of the equation:
x^2 - 22x - 14 = 0.
To complete the square, we take half of the coefficient of x (-22/2 = -11) and square it (-11)^2 = 121. We add this value to both sides of the equation:
x^2 - 22x + 121 - 14 + 121 = 121,
(x - 11)^2 + 107 = 121.
Now, we subtract 107 from both sides of the equation:
(x - 11)^2 = 121 - 107,
(x - 11)^2 = 14.
Taking the square root of both sides, with consideration for both positive and negative square root:
x - 11 = ±√14.
Finally, we add 11 to both sides of the equation, giving us the solutions for x:
x = 11 ± √14.
Therefore, the correct answer is D. x = 11 + √14, 11 - √14.
To complete the square and solve the quadratic equation x^2 - 22x + 1 = 15, we will follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 - 22x + 1 - 15 = 0
Simplifying gives:
x^2 - 22x - 14 = 0
Step 2: Take half of the coefficient of x, square it, and add it to both sides of the equation:
x^2 - 22x + (-22/2)^2 = 14 + (-22/2)^2
Simplifying gives:
x^2 - 22x + 121 = 14 + 121
Step 3: Simplify both sides of the equation:
x^2 - 22x + 121 = 135
Step 4: Rewrite the left side of the equation as a perfect square trinomial:
(x - 11)^2 = 135
Step 5: Take the square root of both sides of the equation, remembering to include the positive and negative square root:
x - 11 = ±√135
Step 6: Simplify the square root of 135:
x - 11 = ±√(9 * 15)
x - 11 = ±3√15
Step 7: Move the constant term to the other side of the equation to solve for x:
x = 11 ± 3√15
Therefore, the solution to the quadratic equation x^2 - 22x + 1 = 15 after completing the square is:
B. x = 11 + 3√15, 11 - 3√15
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 - 22x + 1 = 15
Step 2: Move the 15 to the other side of the equation:
x^2 - 22x + 1 - 15 = 0
x^2 - 22x - 14 = 0
Step 3: Take half of the coefficient of x, square it, and then add it to both sides of the equation. In this case, the coefficient of x is -22, so we take half of it, which is -11. We square this, resulting in 121. Adding 121 to both sides of the equation gives us:
x^2 - 22x + 121 - 14 + 121 = 121
x^2 - 22x + 107 = 121
Step 4: Rewrite the left side of the equation as a perfect square trinomial.
(x - 11)^2 + 107 = 121
Step 5: Move the constant term to the other side of the equation:
(x - 11)^2 = 121 - 107
(x - 11)^2 = 14
Step 6: Take the square root of both sides of the equation:
x - 11 = ±√14
Step 7: Move the 11 to the other side of the equation to isolate x:
x = 11 ± √14
Therefore, the solution to the quadratic equation x^2 - 22x + 1 = 15 is:
x = 11 + √14, 11 - √14
So the correct answer is:
D. x = 11 + √14, 11 - √14