For what values of a is the expression a(4a-16a)/a*|a|+a^2 undefined?
A. a=4
B. a (less than or equal to) 0
C. a=16
D. a (greater than equal to)
Sorry I got to finish D, D should be,
“D. a (greater than or equal to) 0
as written,
a(4a-16a)/a*|a|+a^2 = (4a-16a)*|a|+a^2
for a≠0
The only thing in the denominator is a.
Now, assuming the usual carelessness with parentheses, you might mean
a(4a-16a)/(a*|a|+a^2)
Since |a| = a for a>=0, the denominator is only zero for a=0
|a| = -a for a<0, so the denominator is -a^2+a^2 = 0 for all a<0. So, (B)
In either case, not sure why you wrote 4a-16a, since that's always -12a for any value of a
To determine the values of `a` for which the expression is undefined, we need to identify any values that would result in division by zero or taking the absolute value of a negative number.
Let's analyze the expression step by step:
1. `a(4a - 16a) / (a * |a| + a^2)`
The numerators of both terms (`a(4a - 16a)`) can be simplified:
`a * (-12a) = -12a^2`
The denominator (`a * |a| + a^2`) contains the absolute value of `a`, which can be simplified to:
`|a| = a` if `a >= 0`
`|a| = -a` if `a < 0`
Therefore, we have two separate cases to consider:
Case 1: `a >= 0`
In this case, the expression becomes:
`-12a^2 / (a * a + a^2)`
`-12a^2 / 2a^2`
`= -6`
Case 2: `a < 0`
In this case, the expression becomes:
`-12a^2 / (a * (-a) + a^2)`
`-12a^2 / (-a^2 + a^2)`
`= -12a^2 / 0`
Here, we see that division by zero occurs when `a < 0`, so the expression is undefined for all values of `a` less than zero.
Therefore, the correct answer is:
B. a (less than or equal to) 0
To find the values of "a" for which the expression is undefined, we need to examine the denominator of the expression, which is (a * |a| + a^2).
For an expression to be undefined, the denominator must equal zero, as division by zero is undefined in mathematics.
We can solve the equation (a * |a| + a^2) = 0 to find the values of "a" that make the denominator zero.
Let's break this into two cases:
Case 1: a ≥ 0
When "a" is non-negative, the absolute value of "a" is equal to "a". Thus, the equation becomes:
(a * a + a^2) = 0
2a^2 + a^2 = 0
3a^2 = 0
a^2 = 0
Therefore, in this case, the only value of "a" for which the expression is undefined is a = 0.
Case 2: a < 0
When "a" is negative, the absolute value of "a" is equal to the negation of "a". Thus, the equation becomes:
(a * -a + a^2) = 0
-a^2 + a^2 = 0
0 = 0
In this case, since 0 = 0, the expression is always defined for negative values of "a".
Therefore, the values of "a" for which the expression a(4a-16a)/a*|a|+a^2 is undefined are:
A. a = 4
C. a = 16
Hence, the correct answer is option A and C.