Final Answer:
The value of n that completes the square in the equation x²-12x+n is 144.
Step-by-step explanation:
To complete the square of a quadratic expression, we add and subtract a term inside the parentheses to create a perfect square trinomial. In this case, we want to complete the square of x²-12x.
First, we find half the coefficient of the x term, which is -6. Then, we square this value and add it to the expression:
x²-12x + (-6)^2 = x²-24x + 144
Now, we have a perfect square trinomial, and we can factor it:
(x²-24x+144)^(1/2) = x±12√(n/144)
To find the value of n, we set the exponent inside the parentheses to zero:
0 = (1/2)ln(x±12√(n/144))
This is an equation with two unknowns (x and n). To solve for n, we need to assume a value for x and then calculate n. Let's assume x=0:
0 = (1/2)ln(0±12√(n/144))
Since ln(0) is undefined, this equation doesn't have a solution for x=0. Therefore, we need to assume a different value for x. Let's try x=6:
0 = (1/2)ln(6±12√(n/144))
Now, we can solve for n:
n = 144(6±12√(n/144))^2 = 864 ± 768√(n/144)
To simplify this expression, let's take the square root of both sides:
√n = ±64√(n/144) = ±8√(3n)
Squaring both sides again:
n = 512(3n)^(2/3) = 1792n^(2/3) = 1792^(3/5)*n^(3/5) = 5^6*3^3*n^(3/5) = 5^6*3^3*n^(3) = 5^6*3^3*9^3 = 5^6*3^6 = 7776384000000000000000000000 = 144 (our original guess!) This is an absurdly large number, but it's the only possible value for n that completes the square in our original equation.
So, our final answer is: The value of n that completes the square in the equation x²-12x+n is 144.