what value of q makes the given expression a perfect square?
a^2-8q q
a). 2
b). 4
c). 8
d). 16?
I assume the expression is a^2 - 8q.
Therefore, q = 2 or 8.
To determine the value of q that makes the given expression a perfect square, we need to set it equal to a square of a specific variable.
a^2-8q
Let's set it equal to (a-b)^2, where b is a constant.
(a^2-8q) = (a-b)^2
Expanding the right side:
(a^2 - 8q) = (a^2 - 2ab + b^2)
Comparing the coefficients of similar terms:
-8q = -2ab
To make the expression a perfect square, the coefficient of q should be equal to half the coefficient of ab. Therefore,
-8q = -2ab
Simplifying further:
4q = ab
Since a and b are positive integers, we can determine the value of q by selecting a value for a and b.
Let's try the given options one by one:
a) q = 2
If q = 2, then 4q = 8. However, we cannot determine the value of ab.
b) q = 4
If q = 4, then 4q = 16. We can determine the value of ab by selecting values such as a = 2 and b = 2.
ab = 2 * 2 = 4
Since 4q = ab (16 = 4), this satisfies the condition of a perfect square. Therefore, option (b) q = 4 makes the given expression a perfect square.
To determine the value of q that makes the given expression a perfect square, we need to find a value of q that satisfies the condition.
First, let's write the given expression:
a^2 - 8q
For a quadratic expression to be a perfect square, the coefficient of the linear term (in this case, q) must be equal to half the coefficient of the squared term (in this case, a^2).
So, let's set up the equation:
8q = (1/2) * a^2
Now, let's substitute each answer choice for q and see if it satisfies the equation:
a). q = 2:
8(2) = (1/2) * a^2
16 = (1/2) * a^2
Multiply both sides by 2:
32 = a^2
No value of a can satisfy this equation, so q = 2 does not make the expression a perfect square.
b). q = 4:
8(4) = (1/2) * a^2
32 = (1/2) * a^2
Multiply both sides by 2:
64 = a^2
a = ±8
Substituting a=8 and a=-8 back into the expression:
a^2 - 8q = (8)^2 - 8(4) = 64 - 32 = 32
a^2 - 8q = (-8)^2 - 8(4) = 64 - 32 = 32
Therefore, q = 4 makes the expression a perfect square.
c). q = 8:
8(8) = (1/2) * a^2
64 = (1/2) * a^2
Multiply both sides by 2:
128 = a^2
a = ±√128
We see that √128 cannot be simplified and it is not an integer. Therefore, q = 8 does not make the expression a perfect square.
d). q = 16:
8(16) = (1/2) * a^2
128 = (1/2) * a^2
Multiply both sides by 2:
256 = a^2
a = ±√256
Simplifying, a = ±16
Substituting a=16 and a=-16 back into the expression:
a^2 - 8q = (16)^2 - 8(16) = 256 - 128 = 128
a^2 - 8q = (-16)^2 - 8(16) = 256 - 128 = 128
Therefore, q = 16 also makes the expression a perfect square.
So, the correct answer is (b) 4.