SAT Nonlinear Equations & Systems Question 46 | Free SAT Prep | Aniko

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Question 46·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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Which of the following numbers is a solution to the equation $(x-3)^{2}=25$ ?

For equations of the form $(\text{expression})^2 = \text{number}$ on the SAT, first isolate the squared expression if needed, then take the square root of both sides and remember to include both the positive and negative roots (use $\pm$ ). This gives two simple linear equations to solve; find both possible $x$ -values, then check which ones appear in the answer choices and, if needed, quickly verify by substitution into the original equation. This approach is fast and helps you avoid the common mistake of forgetting one of the solutions.

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Focus on the squared expression

Look at the expression inside the square: $x-3$ . Think about what values $x-3$ could have so that its square equals 25.

Use both positive and negative square roots

When you take the square root of both sides of $(x-3)^2=25$ , remember that both a positive and a negative number can square to 25.

Solve the resulting simple equations

After you write $x-3$ equal to each square root, solve each resulting equation for $x$ and then see which of those values appears in the answer choices.

Check by substitution

If you think a choice might work, plug it into $(x-3)^2$ and see if the result is 25.

Desmos Guide

Graph both sides of the equation

In Desmos, type y = (x - 3)^2 on one line and y = 25 on another line. This graphs the parabola and the horizontal line.

Find the intersection points

Look for the points where the parabola and the horizontal line intersect. Click on each intersection; Desmos will show the coordinates $(x, y)$ of these points.

Compare x-values to the answer choices

Note the $x$ -coordinates of the intersection points (these are the solutions to the equation). Compare those $x$ -values to the choices 5, 7, 9, and $-2$ , and see which one matches.

Step-by-step Explanation

Understand what the equation is saying

The equation is

(x-3)^2 = 25

This means the quantity $x-3$ is being squared (multiplied by itself) and the result is 25. We need all values of $x$ that make this true.

Undo the square with square roots (remember both signs)

To undo a square, take the square root of both sides. Because squaring a positive or a negative number can give the same result, you must use both the positive and negative square roots:

(x-3)^2 = 25 \\[0.7em] \Rightarrow x-3 = \pm 5

So there are two equations to solve:

$x-3 = 5$
$x-3 = -5$

Solve each simple equation for x (symbolically)

Solve each of the two linear equations by adding 3 to both sides.

From $x-3=5$ :

x = 5 + 3

From $x-3=-5$ :

x = -5 + 3

These give two possible values of $x$ .

Find the numerical values and match to the choices

Now compute the two values:

From $x = 5 + 3$ , we get $x = 8$ .
From $x = -5 + 3$ , we get $x = -2$ .

So the equation has two solutions: $x=8$ and $x=-2$ .

Look at the answer choices: 5, 7, 9, and $-2$ . Only $-2$ is one of the solutions we found.

To confirm, substitute $x=-2$ into the original equation:

(-2-3)^2 = (-5)^2 = 25

The equation is true, so the correct answer is $-2$ (choice D).

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation