Question 78·Medium·Advanced Math

Type $y=(3x-4)^2+5(x+2)-(x^2-16)$. Desmos draws a U-shaped curve. The $y$ on the left is the graph’s output; every $y$ from the problem becomes $x$ on the right so Desmos can graph it.

Add $y=8x^2-29x+42$. Its curve meets the original at $x=0$ but separates from it elsewhere. One shared point isn’t a match: equivalent expressions must agree at every input.

Add $y=8x^2-19x+26$. Desmos draws it below the original rather than on top of it, so these are different curves.

Add $y=7x^2-19x+42$. Its curve meets the original at $x=0$ but pulls away as $x$ moves from $0$. Again, meeting once isn’t enough.

Add $y=8x^2-19x+42$. Its curve sits directly on the original throughout the window. Hide and show this last line to check that the curve doesn’t move. The equivalent expression is $8y^2-19y+42$. Choice D.

No Desmos needed. Algebra checks the match for every value of $y$. A square means multiplying the expression by itself. Multiply each term in the first copy by each term in the second, keeping both middle products:

Add the two terms containing $y$:

The $-24y$ comes from two copies of $-12y$.

The $5$ multiplies both terms inside its parentheses:

Subtracting the last parentheses means multiplying each term by $-1$, so both signs change:

Replace each part with the form you found, keeping the $+16$ from the subtraction:

Like terms have the same variable part. Group the $y^2$ terms, the $y$ terms, and the numbers separately:

Combine those three groups:

So the expression is equivalent to $8y^2-19y+42$. Choice D.