Given that 9x²+36x-40+k is a perfect square,find the value of k
To determine whether the given expression is a perfect square, we can use the following steps:
1. Rewrite the given expression in the form of "(ax + b)²".
2. Identify the values of 'a' and 'b'.
3. If the expression can be rewritten in the form of "(ax + b)²", then the value of 'k' can be determined as 'b²'.
Now, let's follow these steps to find the value of 'k':
1. Rewrite the given expression in the form of "(ax + b)²":
9x² + 36x - 40 + k = (ax + b)²
We need to determine the values of 'a' and 'b'. For a perfect square, 'a' should be equal to the square root of the coefficient of x², and 'b' should be equal to half of the coefficient of x.
Comparing coefficients, we have:
a² = 9
2ab = 36
2. Solving the equations:
Taking the square root of both sides, we find a = ±3 (since the square root of 9 can be either +3 or -3).
Substituting the value of 'a' in the second equation:
2(±3)b = 36
±6b = 36
b = ±6
So we have two possible cases:
Case 1: When a = 3 and b = 6
Case 2: When a = -3 and b = -6
3. Determining the value of 'k':
We can determine the value of 'k' using 'b²' from each case:
Case 1: k = b² = 6² = 36
Case 2: k = b² = (-6)² = 36
Therefore, the value of 'k' is 36.
To determine the value of k such that the expression 9x² + 36x - 40 + k is a perfect square, we need to complete the square.
1. First, let's factor out the common factor of 9 from the first two terms:
9x² + 36x = 9(x² + 4x)
2. To complete the square, we need to take half of the coefficient of the x term (which is 4 in this case), square it, and add it to both sides of the equation.
Half of 4 is 2, and when squared it becomes 4.
Adding 4 to both sides:
9(x² + 4x + 4) = 9(x + 2)²
3. Now, we can express the original expression as a perfect square:
9(x + 2)² - 40 + k
At this point, for the expression to be a perfect square, the constant term (- 40 + k) should be a perfect square as well.
4. To achieve this, we look for the square root of 40. The square root of 40 is approximately 6.324.
To make (- 40 + k) a perfect square, we need to add the square of half that value to it:
(- 40 + 6.324²) = (- 40 + 40.0039) = 0.0039
So, the value of k that makes the expression a perfect square is approximately 0.0039.
Therefore, the value of k is approximately 0.0039.