SAT Nonlinear Equations & Systems Question 192 | Free SAT Prep | Aniko

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Question 192·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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y = (x-2)^2 + 2\\[0.7em] (x-2)^2 + (y-3)^2 = 1

How many solutions are there to the system of equations above?

For systems with one equation already solved for a variable (like $y = \text{expression}$ ), use substitution: plug that expression into the other equation to get a single-variable equation. When you see repeated complicated pieces such as $(x-2)^2$ , temporarily replace them with a simpler variable (like $t$ ), solve the resulting quadratic, and then translate each $t$ -value back to possible $x$ -values. Finally, for each $x$ , compute $y$ and count all distinct ordered pairs—being careful that equations like $(x-a)^2 = k$ can yield one or two real $x$ -values depending on $k$ . This approach keeps the algebra organized and minimizes mistakes under time pressure.

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Notice the structure of the system

One equation already has $y$ isolated. How can you use that to combine the two equations into one equation in a single variable or expression?

Substitute for y in the circle equation

Replace $y$ in $(x-2)^2 + (y-3)^2 = 1$ with the expression from $y = (x-2)^2 + 2$ , and then simplify. Be careful with the parentheses when you subtract 3.

Use a temporary variable to simplify

After substitution, you will see repeated $(x-2)^2$ terms. Let $t = (x-2)^2$ to get an equation involving $t$ only, then solve it.

Translate each t-value back to x and y

For each value of $t$ that you find, solve $(x-2)^2 = t$ for $x$ , then plug each $x$ into $y = (x-2)^2 + 2$ to find corresponding $y$ -values. Count all distinct ordered pairs you get.

Desmos Guide

Graph the parabola

Type y = (x-2)^2 + 2 into Desmos. This will display the upward-opening parabola with vertex at $(2,2)$ .

Graph the circle

On a new line, type (x-2)^2 + (y-3)^2 = 1. Desmos can graph this implicit equation as a circle with center $(2,3)$ and radius $1$ .

Find the intersection points

Zoom in or adjust the view so you can clearly see where the circle and parabola meet. Tap or click on each intersection point Desmos shows; it will display the coordinates. Count how many distinct intersection points there are—this count is the number of solutions to the system.

Step-by-step Explanation

Interpret the equations geometrically (optional check)

$y = (x-2)^2 + 2$ is a parabola that opens upward with vertex at $(2,2)$ .
$(x-2)^2 + (y-3)^2 = 1$ is a circle with center $(2,3)$ and radius $1$ .
We are looking for the intersection points of this parabola and this circle, i.e., the ordered pairs $(x,y)$ that satisfy both equations at the same time.

Use substitution to combine the equations

From the first equation,

y = (x-2)^2 + 2.

Substitute this expression for $y$ into the second equation:

(x-2)^2 + (y-3)^2 = 1 \\[0.7em] (x-2)^2 + \big((x-2)^2 + 2 - 3\big)^2 = 1.

Simplify inside the parentheses:

(x-2)^2 + \big((x-2)^2 - 1\big)^2 = 1.

Simplify with a substitution for the squared term

Let $t = (x-2)^2$ . Then the equation becomes

t + (t - 1)^2 = 1.

Now expand $(t-1)^2$ :

t + (t^2 - 2t + 1) = 1.

Combine like terms:

t^2 - t + 1 = 1.

Subtract $1$ from both sides:

t^2 - t = 0.

Factor:

t(t - 1) = 0.

So $t$ can be two possible values, which you will now convert back to information about $x$ and $y$ . Remember $t = (x-2)^2$ .

Find all possible x-values and corresponding y-values

From $t(t-1) = 0$ , we have $t = 0$ or $t = 1$ .

Case 1: $t = 0$

Then $(x-2)^2 = 0$ , so $x - 2 = 0$ and $x = 2$ (only one value, because the square root of $0$ is just $0$ ).
Use $y = (x-2)^2 + 2$ : when $x = 2$ , we get $y = 0 + 2 = 2$ .
This gives one solution: $(2,2)$ .

Case 2: $t = 1$

Then $(x-2)^2 = 1$ , so $x - 2 = 1$ or $x - 2 = -1$ .
Therefore $x = 3$ or $x = 1$ .
For both $x$ -values, $t = 1$ , so $y = t + 2 = 1 + 2 = 3$ .
This gives two more solutions: $(3,3)$ and $(1,3)$ .

Altogether there are three distinct ordered pairs $(x,y)$ that satisfy the system, so there are exactly 3 solutions.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation