Determine the value or values of d that complete the square for the expression.
(a) x^2 + dx +64
(b) x^2 + dx +49/4
that last number = (d/2)^2
so
64 = d^2/4
d^2 = 4 * 64
d = 2*8
d = 16 like (x+8)(x+8)
then the second one
d^2/4 = 49/4
so d = 7
like (x+7/2)(x+7/2) = x^2 + 7 x + 49/4
Lee
To complete the square for expressions of the form x^2 + dx + c, we need to find a value or values of d that make the expression a perfect square trinomial.
(a) For the expression x^2 + dx + 64
Step 1: Take half of the coefficient of x and square it.
Half of d is d/2, so we have (d/2)^2 = d^2/4.
Step 2: Add the value obtained in Step 1 to the expression.
x^2 + dx + 64 + d^2/4
Now we have a perfect square trinomial: (x + d/2)^2 = x^2 + dx + d^2/4.
Therefore, the value of d that completes the square for the expression x^2 + dx + 64 is d = ± 8.
(b) For the expression x^2 + dx + 49/4
Step 1: Take half of the coefficient of x and square it.
Half of d is d/2, so we have (d/2)^2 = d^2/4.
Step 2: Add the value obtained in Step 1 to the expression.
x^2 + dx + 49/4 + d^2/4
Now we have a perfect square trinomial: (x + d/2)^2 = x^2 + dx + (d^2/4 + 49/4) = x^2 + dx + (d^2 + 49)/4.
Therefore, the value of d that completes the square for the expression x^2 + dx + 49/4 is any real number d.
To complete the square for the expression, we need to find the value of "d" that makes the expression a perfect square trinomial.
(a) x^2 + dx + 64:
To complete the square, we need to add a constant term to the expression. In this case, since the coefficient of x is "d," we want to add (d/2)^2 to make it a perfect square trinomial.
So, adding (d/2)^2 to x^2 + dx + 64, we get:
x^2 + dx + (d/2)^2 + 64
For it to be a perfect square trinomial, the constant term should be equal to the square of half the coefficient of the "x" term. Therefore, we have:
(d/2)^2 = 64
Taking the square root of both sides, we get:
d/2 = ±√64 = ±8
Multiplying by 2 to both sides, we find:
d = ±16
Therefore, the values of "d" that complete the square for the expression x^2 + dx + 64 are d = 16 and d = -16.
(b) x^2 + dx + 49/4:
Similarly, for this expression, we want to add a constant term to make it a perfect square trinomial. In this case, we add (d/2)^2 to x^2 + dx + 49/4.
x^2 + dx + (d/2)^2 + 49/4
For it to be a perfect square trinomial, we want the constant term to be equal to the square of half the coefficient of the "x" term. Therefore, we have:
(d/2)^2 = 49/4
Taking the square root of both sides, we get:
d/2 = ±√(49/4) = ±7/2
Multiplying by 2 to both sides, we find:
d = ±7
Therefore, the values of "d" that complete the square for the expression x^2 + dx + 49/4 are d = 7/2 and d = -7/2.