After you substitute and set the expressions for $y$ equal, you should expand $(x - 3)^2$ and collect like terms. Do you end up with a quadratic equation of the form $ax^2 + bx + c = 0$?

Use the quadratic formula on your quadratic equation. Once you get expressions for $x$ involving $\sqrt{13}$, compare them carefully to each answer choice, paying attention to both the numerator and the denominator.

In Desmos, enter the two equations as

• y = (x - 3)^2 + 1
• y = 7 - x

This will plot a parabola and a line on the same coordinate plane.

Use Desmos’s intersection feature (click or tap where the graphs cross, or tap on one graph and select the intersection) to see the coordinates of the intersection points. Note the $x$-values of these intersection points, since those are the $x$-solutions of the system.

Type each answer choice into Desmos as a separate expression to see its decimal value, and compare these values to the $x$-coordinates of the intersection points from step 2. The choice whose value matches one of the intersection $x$-values is the correct answer.

The system is

Since both equations involve $y$, write the second equation as $y = 7 - x$ and set the two expressions for $y$ equal:

Now you have a single equation in terms of $x$.

First expand $(x - 3)^2$:

Substitute this into the equation:

Combine like terms on the left:

Move everything to one side to put the equation in standard quadratic form $ax^2 + bx + c = 0$:

So the equation you need to solve is $x^2 - 5x + 3 = 0$.

For the quadratic equation $x^2 - 5x + 3 = 0$, identify $a$, $b$, and $c$:

• $a = 1$
• $b = -5$
• $c = 3$

The quadratic formula is

Compute the discriminant $b^2 - 4ac$:

So the solutions will involve $\sqrt{13}$.

Substitute $a = 1$, $b = -5$, and the discriminant $13$ into the quadratic formula:

So the two $x$-values that satisfy the system are

The question asks which of the answer choices is a solution for $x$; among the options, only $\dfrac{5+\sqrt{13}}{2}$ appears, so the correct choice is $\dfrac{5+\sqrt{13}}{2}$.