Math

Question Can 4x(x3)-4 x(-x-3) ever be negative? Choose the best: No, 4x24 x^{2} and 12x12 x are positive; Yes, 4x2+12x4 x^{2}+12 x is negative when xx is between -3 and 0; No, both factors are negative, so the product must be positive.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression 4x(x3)-4x(-x-3).
2. We need to determine if this expression can ever be negative.
3. We will analyze the expression by expanding it and looking at the signs of the resulting terms.

STEP 2

First, let's expand the expression to see what terms we have.
4x(x3)=4x2+12x-4x(-x-3) = 4x^2 + 12x

STEP 3

Now, we will analyze the terms in the expanded expression.
1. The term 4x24x^2 is always non-negative because the square of any real number, whether positive or negative, is non-negative.
2. The term 12x12x can be positive, negative, or zero, depending on the value of xx.

STEP 4

Since 4x24x^2 is always non-negative, the sign of the expression 4x2+12x4x^2 + 12x depends on the term 12x12x.

STEP 5

Let's consider the term 12x12x:
1. If x>0x > 0, then 12x>012x > 0.
2. If x<0x < 0, then 12x<012x < 0.
3. If x=0x = 0, then 12x=012x = 0.

STEP 6

We need to find out if there is any value of xx for which the entire expression 4x2+12x4x^2 + 12x is negative.

STEP 7

Since 4x24x^2 is always non-negative, the only way for 4x2+12x4x^2 + 12x to be negative is if 12x12x is negative and its absolute value is greater than the value of 4x24x^2.

STEP 8

Let's find the values of xx for which 12x12x is negative:
12x<0x<012x < 0 \Rightarrow x < 0

STEP 9

Now, we need to check if there is any value of xx less than 0 for which 12x>4x2|12x| > 4x^2.

STEP 10

Let's set up the inequality to find such values of xx:
12x>4x2|12x| > 4x^2

STEP 11

Since we are looking for x<0x < 0, we can remove the absolute value and change the sign of 12x12x:
12x>4x2-12x > 4x^2

STEP 12

Divide both sides of the inequality by 4x4x (noting that x0x \neq 0 to avoid division by zero):
3>x-3 > x

STEP 13

This inequality tells us that for x<3x < -3, the expression 4x2+12x4x^2 + 12x is negative.

STEP 14

However, we must remember that 4x24x^2 is non-negative, and it grows faster than 12x|12x| as xx becomes more negative. This means that there is no value of xx for which 4x24x^2 is smaller than 12x|12x|.

STEP 15

Therefore, the expression 4x2+12x4x^2 + 12x can never be negative because the positive term 4x24x^2 will always be greater than or equal to the absolute value of the negative term 12x12x for any real number xx.

STEP 16

The correct explanation is: No, because both factors are negative, so the product must be positive. The expression 4x(x3)-4x(-x-3) simplifies to 4x2+12x4x^2 + 12x, which cannot be negative because 4x24x^2 is always non-negative and will always be greater than or equal to the absolute value of 12x12x for any real number xx.

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