Question 5·200 Super-Hard SAT Math Questions·Advanced Math

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Consider the quadratic expression

ax^2 + 130x + c

where $a$ and $c$ are positive constants. Suppose the expression can be written in the form $(jx+k)(kx+j)$ , where $j$ and $k$ are positive constants.

Which choice gives the greatest possible value of $ac$ ?

Factors that swap their coefficients signal a polynomial identity: expanding shows that the first and last coefficients match, while the middle coefficient is a sum of two squares. Match those coefficients, then use a nonnegative square to bound the product you need. The tempting slip is to treat the fixed sum of squares as the product itself.

Hints

Hint 1
A coefficient is the number multiplying a power of $x$ . In the product, the $x^2$ term comes from $(jx)(kx)$ , and the constant comes from $k\cdot j$ . What does that tell you about both $a$ and $c$ ?
Hint 2
The middle term has two sources: $(jx)j$ and $k(kx)$ . Their coefficients add to $j^2+k^2$ , which must match the coefficient of $x$ in the given expression. How could you relate that sum to $2jk$ ?
Hint 3
A square cannot be negative, so $(j-k)^2\ge0$ . Expand it to bound $jk$ , then check when equality happens. The question asks for the greatest possible value, not merely a possible one.

Step-by-step

Match coefficients and bound the product
Step 1
Expand the swapped factors
No Desmos needed. Coefficient matching and a nonnegative square give an exact maximum. Multiply the factors, keeping the two middle products separate:
$(jx+k)(kx+j)=jkx^2+j^2x+k^2x+jk.$
Combine the $x$ terms:
$(jx+k)(kx+j)=jkx^2+(j^2+k^2)x+jk.$
Step 2
Match coefficients
The two forms describe the same expression for every value of $x$ , so their coefficients, the numbers multiplying matching powers of $x$ , must agree:
$a=jk,\qquad j^2+k^2=130,\qquad c=jk.$
Step 3
Rewrite the product the question asks for
Since both $a$ and $c$ equal $jk$ , multiply them:
$ac=(jk)(jk)=(jk)^2.$
Because $j$ and $k$ are positive, making $jk$ as large as possible also makes $ac$ as large as possible.
Step 4
Bound the product using a square
A square cannot be negative. Expand $(j-k)^2$ and use the fixed sum $j^2+k^2=130$ :
$0\le(j-k)^2=j^2-2jk+k^2=130-2jk.$
Add $2jk$ to both sides:
$2jk\le130.$
Divide by $2$ :
$jk\le65.$
With a fixed sum of squares, the product is greatest when the two numbers are equal.
Step 5
Check that the bound is possible
Equality in the square bound requires $j=k$ . Substitute $k=j$ into $j^2+k^2=130$ :
$2j^2=130.$
Divide by $2$ :
$j^2=65.$
Take the positive square root, as the problem requires positive constants:
$j=k=\sqrt{65}.$
These values satisfy the given middle coefficient, so the bound can be reached.
Step 6
Find the greatest coefficient product
At that bound, $jk=65$ . Square it to get the requested product:
$ac=(jk)^2=65^2=4225.$
So the greatest possible value of $ac$ is $4225$ . Choice C.

Question 5·200 Super-Hard SAT Math Questions·Advanced Math

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