M7MBA wrote: ↑

Thu Dec 17, 2020 11:35 am

If \(x > 0,\) what is the least possible value of \(-2\sqrt{5x} + x + 9 ?\)

A. \(0\)
B. \(1\)
C. \(\sqrt5\)
D. \(4\)
E. \(9\)

Answer: D

Solution:

If x = 5, we have -2√(25) + 5 + 9 = -10 + 14 = 4. Now let’s see if the expression can be less than 4. If it can, we have:

-2√(5x) + x + 9 < 4

-2√(5x) < -x - 5

2√(5x) > x + 5

Since x is positive, both sides are positive. So if we square both sides of the inequality, we have:

4(5x) > x^2 + 10x + 25

0 > x^2 - 10x + 25

0 > (x - 5)^2

However, since (x - 5)^2 is a perfect square, it’s non-negative, so 0 can’t be greater than (x - 5)^2. In other words, it’s impossible to have the expression -2√(5x) + x + 9 be less than 4. Therefore, 4 is the least possible value of the expression.

Alternate Solution:

Notice that x is the square of √x and 2√(5x) is twice the product of √x and √5. Thus, we can write the given expression as follows:

x - 2√(5x) + 9

x - 2√(5x) + 5 + 4

(√x)^2 - 2(√x)(√5) + (√5)^2 + 4

(√x - √5)^2 + 4

We note that the square of some expression is always nonnegative, thus the smallest possible value of (√x - √5)^2 is 0. It follows that the smallest possible value of x - 2√(5x) + 9 is 4.

Answer: D