Question 1009·Hard·Math

Type $2y-4x=-18$ and $x^2-y=29$ on separate Desmos lines. Click both intersections. Desmos shows approximately $(-3.583,-16.165)$ and $(5.583,2.165)$. Each point lies on both graphs, so each first coordinate is a possible $x$. These clicked coordinates are rounded, not exact answers.

Type $1-\sqrt{21}$ on a new line. Desmos shows approximately $-3.583$, matching the $x$-coordinate of the left intersection. So $1-\sqrt{21}$ is one possible value of $x$. Choice B.

Add $4x$ to both sides of the line’s equation:

Divide both sides by $2$: $y=2x-9$. Now the line gives a $y$-value for any $x$.

Substitution means replacing a variable with an equal expression. Replace $y$ in $x^2-y=29$ with $2x-9$: $x^2-(2x-9)=29$. At an intersection, both equations must give the same $y$.

Distribute the minus sign across both terms inside the parentheses:

Subtract $9$ from both sides to leave the $x$ terms together:

To complete the square, take half of the $x$ coefficient, $-2$, and square it: $(-1)^2=1$. Add $1$ to both sides so the left side can become a squared expression:

Since $(x-1)^2=x^2-2x+1$, rewrite the left side:

Take square roots of both sides. A number and its opposite have the same square, so keep both signs:

Add $1$ to both sides: $x=1\pm\sqrt{21}$. The minus sign gives one possible value, $1-\sqrt{21}$. Choice B.