If p 0, for what value of p is the expression, p

Question

If p > 0, for what value of p is the expression,  −p + 10\sqrt{p}- 32  the greatest?

(A)  \sqrt{5}

(B) 3

(C) 5

(D) 12

(E) 25

Bunuel · Accepted Answer

Let's rewrite the given expression and complete the square:

−p + 10\sqrt{p}- 32=

=−(p - 10\sqrt{p} + 25)- 7=

=−(\sqrt{p} - 5)^2 - 7 .

Since the square of a number is positive or 0,  −(\sqrt{p} - 5)^2  is negative or 0. Hence, to maximize  −(\sqrt{p} - 5)^2 - 7 , the term  −(\sqrt{p} - 5)^2  must be 0, which means √p must be 5. Therefore, p = 5^2, which leads to p = 25.

Answer: E.

ArinK2101 · Answer

why is it that you took out the 7? i don't get how you mathematically solved this, because if you wanted a perfect square for example, you could have taken out -16...so you get 32-16=16(also a square)

Krunaal · Answer

We try to equate,  −p + 10\sqrt{p}- 32=  to  a^2 - 2ab + b^2

=>  −(p - 10\sqrt{p} + 32)=  to  a^2 - 2ab + b^2

We can see that  a  can be  \sqrt{p}  and  2ab = 10\sqrt{p}

=>  2*\sqrt{p}*b = 10\sqrt{p}

=> b = 5

Now if  a  =  \sqrt{p}  and  b = 5

We'll have  b^2  =  25 , thus we take out  -7

If it were  b = 4 , then we could've thought of  16  and taken out  16 .

Hope it helps.

If p > 0, for what value of p is the expression, p

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