Determine the value or values of d that complete the square for the expression. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
(a)
x^2 + 7x + d
d =
b)
x^2 − 18x + d
d =
(c)
x^2 + dx + 64
d =
(d)
x^2 + dx + 49/4
d =
(7/2)^2
(18/2)^2
(√64)*2
(√49/4)*2
Note that (a) and (d) are the same polynomial
(a) The value of d that completes the square for x^2 + 7x + d is 24. Why? Because when you factor the first two terms, you get (x + 3.5)^2, and the value of d is the constant term in that expression, which is 24. So, d = 24.
(b) The value of d that completes the square for x^2 - 18x + d is 81. Are you surprised? Well, the coefficient of x is -18, which means we have to take half of that to complete the square. Half of -18 is -9, and when you square -9, you get 81. Thus, d = 81.
(c) The value of d that completes the square for x^2 + dx + 64 is -d. Wait, what? Yes, that's right! When you factor the first two terms, you get (x + 0.5d)^2, and the value of d is the negative of the constant term in that expression, which is -64. Therefore, d = -64.
(d) The value of d that completes the square for x^2 + dx + 49/4 is 7/2. How did I come up with that? Simple! The constant term in the factored expression (x + 0.5d)^2 is (0.5d)^2, and we know that (0.5d)^2 equals (d/2)^2. Equating this to 49/4, we get (d/2)^2 = 49/4, which implies d/2 = 7/2. Solving for d, we find that d = 7/2.
So, the values of d that complete the square for the expressions are:
(a) d = 24
(b) d = 81
(c) d = -64
(d) d = 7/2
(a) To complete the square for the expression x^2 + 7x + d, we need to take half of the coefficient of x, square it, and add it to the expression.
The coefficient of x is 7, so half of it is 7/2, and when squared, it is (7/2)^2 = 49/4.
Therefore, d = 49/4.
(b) For the expression x^2 -18x + d, we complete the square by taking half of the coefficient of x, squaring it, and adding it to the expression.
The coefficient of x is -18, so half of it is -18/2 = -9, and when squared, it is (-9)^2 = 81.
Therefore, d = 81.
(c) To complete the square for the expression x^2 + dx + 64, we need to find the value of d that satisfies the equation.
In order to complete the square, we take half of the coefficient of x, square it, and add it to the expression. However, since the exact value of d is not given, we cannot determine a specific value.
(d) Similarly, to complete the square for the expression x^2 + dx + 49/4, we need to find the value of d that satisfies the equation.
Taking half of the coefficient of x, squaring it, and adding it to the expression, we get the following equation:
(d/2)^2 + 49/4 = 0.
We can solve this equation for d by subtracting 49/4 from both sides and then taking the square root of both sides, but note that both solutions will be the negative of each other.
Therefore, d = -49/2 or d = -24.5.
To complete the square for the given expressions, we need to find the value(s) of "d" that will make the quadratic expression a perfect square trinomial.
(a) For the expression x^2 + 7x + d, we can complete the square by taking half of the coefficient of x, squaring it, and adding it to both sides of the equation. Half of 7 is 3.5, and squaring it gives us 12.25. Therefore, we add 12.25 to both sides of the equation:
x^2 + 7x + 12.25 + d = x^2 + 7x + 12.25
This forms a perfect square trinomial since the left side can be factored into (x + 3.5)^2. So the value of d that completes the square is 12.25.
(a) -> d = 12.25
(b) For the expression x^2 - 18x + d, we again take half of the coefficient of x, which is -18 in this case, square it, and add it to both sides of the equation. Half of -18 is -9, and squaring it gives us 81. Adding 81 to both sides, we get:
x^2 - 18x + 81 + d = x^2 - 18x + 81
This forms a perfect square trinomial, which can be factored as (x - 9)^2. Therefore, the value of d that completes the square is 81.
(b) -> d = 81
(c) For the expression x^2 + dx + 64, we don't have a constant term, and we need to complete the square by taking half of the coefficient of x, squaring it, and adding it to both sides. Therefore, we take half of 'd' and square it, resulting in d^2/4. The equation becomes:
x^2 + dx + d^2/4 + 64 = x^2 + dx + 64
This expression can't be factored into a perfect square trinomial since there is still the term d^2/4 present. Therefore, there is no value of 'd' that completes the square for this expression.
(c) -> NO SOLUTION
(d) For the expression x^2 + dx + 49/4, we take half of the coefficient of x, which is 'd/2', and square it, giving us (d/2)^2. Adding it to both sides of the equation, we get:
x^2 + dx + (d/2)^2 + 49/4 = x^2 + dx + 49/4
This forms a perfect square trinomial, which can be factored as (x + d/2)^2. Therefore, the value of d that completes the square is d = 2.
(d) -> d = 2