(a) Complete the square by writing in the form . Note: the numbers and can be positive or negative.
x^2-14x-17=(x-7)^2+
(b) Solve the equation by completing the square or using the quadratic formula. If there is more than one correct answer, enter your answers as a comma separated list. If there are no solutions, enter NONE.
x^2-14x-17=0
I keep getting this wrong but I don't know why.help please
x^2 - 14x + .... = 17 + ....
x^2 - 14x + 49 = 17+ 49
(x-7)^2 = 66
x-7 = ± √66
x = 7 ± √66
by formula:
x = (14 ± √(14^2 - 4(1)(-17))/2
= (14 ± √264)/2
= (14 ± 2√66)/2
= 7 ± √66
When the coefficient of x^2 is 1 and the middle term is even, I always use "completing the square" rather than the formula. Notice you get the answer already in simplified form
To complete the square and solve the equation x^2 - 14x - 17 = 0, follow these steps:
(a) Completing the Square:
Step 1: Move the constant term (-17) to the right side of the equation by adding 17 to both sides:
x^2 - 14x = 17
Step 2: Take half of the coefficient of x (-14 in this case) and square it. So, (-14/2)^2 = 49.
Add 49 to both sides of the equation:
x^2 - 14x + 49 = 17 + 49
Step 3: Simplify and rewrite the equation:
(x - 7)^2 = 66
Hence, by completing the square, the equation becomes (x - 7)^2 = 66.
(b) Solving the Equation:
Step 4: Take the square root of both sides of the equation:
√[(x - 7)^2] = ±√66
Step 5: Solve for x by considering both the positive and negative square roots:
(x - 7) = ±√66
For the positive square root:
x - 7 = √66
x = 7 + √66
x = 7 + √(2 * 33)
x = 7 + √(2 * 3^2 * 11)
x = 7 + 3√22
For the negative square root:
x - 7 = -√66
x = 7 - √66
x = 7 - √(2 * 3^2 * 11)
x = 7 - 3√22
Therefore, the solutions to the equation x^2 - 14x - 17 = 0 are:
x = 7 + 3√22, 7 - 3√22.