Question 179·200 Super-Hard SAT Math Questions·Advanced Math

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The real numbers $x$ and $y$ satisfy the system

\begin{cases} \sqrt{x+2}-\sqrt{y+2}=1 \\ x+y=14 \end{cases}

Which choice is the value of $xy$ ?

A nonlinear system has two equations in the same unknowns, with at least one equation that isn't a line. Graph both original equations in Desmos and click where they meet. The intersection gives you the coordinates, but you still have to calculate what the question asks for. Desmos rounds point labels, so match the approximate product to an exact choice rather than treating the displayed decimals as exact.

Hints

Hint 1
A solution to a system makes both equations true. Graph each equation as written, then click the point where their graphs meet to read $x$ and $y$ .
Hint 2
The question asks for $xy$ , which means $x$ multiplied by $y$ . Use the two coordinates from the intersection, not either coordinate alone.
Hint 3
A graph's point label is rounded, so its product may be a little off. Evaluate the fraction choices in Desmos and look for the one that matches that approximate product.

Step-by-step

Approach 1: Graph the system and match the product
Step 1
Read the solution from the intersection
Type both equations as written into Desmos. Click their intersection, near $(9.958,4.042)$ . A point on both graphs makes both equations true, so this gives $x\approx9.958$ and $y\approx4.042$ .
Step 2
Multiply the coordinates
The requested product $xy$ means multiply the two coordinates. Type $9.958\cdot4.042$ ; Desmos shows $40.250236$ . This is approximate because the coordinates you clicked were rounded.
Step 3
Match the exact choice
Type the fraction choices on separate lines. Desmos shows $40.25$ , $72.25$ , $24.25$ , and $-40.25$ . Only $40.25$ matches the approximate product; the small difference comes from the rounded graph label. So $xy=\frac{161}{4}$ . Choice A.

Approach 2: Find the exact product without finding x and y
Step 1
Square the entire difference
For an exact route, square the whole left side, not each square root separately. Since its value is $1$ :
$\left(\sqrt{x+2}-\sqrt{y+2}\right)^2=1^2.$
Step 2
Keep the middle term
Use $(a-b)^2=a^2+b^2-2ab$ to expand the squared difference. The product of the roots is the middle term:
$(x+2)+(y+2)-2\sqrt{(x+2)(y+2)}=1.$
Step 3
Use the given sum
The two terms outside the root add to $(x+y)+4$ . Substitute $x+y=14$ :
$14+4-2\sqrt{(x+2)(y+2)}=1.$
Add $14$ and $4$ :
$18-2\sqrt{(x+2)(y+2)}=1.$
Step 4
Isolate the remaining square root
Get the square root alone before squaring again. Subtract $18$ :
$-2\sqrt{(x+2)(y+2)}=-17.$
Divide by $-2$ :
$\sqrt{(x+2)(y+2)}=\frac{17}{2}.$
Step 5
Find the shifted product
Square both sides to clear the remaining root:
$(x+2)(y+2)=\left(\frac{17}{2}\right)^2=\frac{289}{4}.$
This is the product of $x+2$ and $y+2$ , not yet $xy$ .
Step 6
Expose xy inside the product
Expand the shifted product so you can see the requested $xy$ :
$xy+2x+2y+4=\frac{289}{4}.$
Factor $2$ from the middle terms:
$xy+2(x+y)+4=\frac{289}{4}.$
Step 7
Replace the sum
Use $x+y=14$ again in the expanded product:
$xy+2(14)+4=\frac{289}{4}.$
Step 8
Calculate the requested product
Subtract the terms added to $xy$ :
$xy=\frac{289}{4}-2(14)-4.$
Type that expression into Desmos. It shows $40.25$ ; its fraction button gives $\frac{161}{4}$ . So the exact value of $xy$ is $\frac{161}{4}$ . Choice A.

Question 179·200 Super-Hard SAT Math Questions·Advanced Math

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