The positive real numbers , , and satisfy Which equation correctly expresses in terms of and ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 231 | Free SAT Prep | Aniko

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Question 231·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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The positive real numbers $a$ , $b$ , and $c$ satisfy

a = \frac{b}{c^2 - 4}.

Which equation correctly expresses $c$ in terms of $a$ and $b$ ?

When a variable is in the denominator and squared, treat the other symbols as constants and focus on isolating the variable step by step: first clear the denominator by multiplying both sides, then isolate the squared term, and finally take the square root, remembering to use the sign information given in the problem (such as “positive real numbers”) to choose the correct root. This systematic approach avoids algebra mistakes with fractions and signs and quickly leads you to the correct expression in terms of the other variables.

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Remove the denominator

How can you get rid of the denominator $c^2 - 4$ in the equation $a = \dfrac{b}{c^2 - 4}$ ? Think about multiplying both sides by that denominator.

Get an expression for $c^2$

Once you have an equation like $a(c^2 - 4) = b$ , what can you do to isolate $c^2 - 4$ and then $c^2$ ? Consider dividing by $a$ and then adding or subtracting a constant.

Go from $c^2$ to $c$

After you have an equation of the form $c^2 = \text{(expression in a and b)}$ , what operation lets you solve for $c$ ? Remember that taking a square root usually gives two possible values—how does the condition that $c$ is positive affect your choice?

Desmos Guide

Choose test values for $a$ and $b$

In Desmos, define $a$ and $b$ as positive numbers or sliders (for example, type a = 1 and b = 5, or create sliders). Make sure they are positive as the problem states.

Compute $c$ from each candidate formula

For each of the four answer choices, enter an expression like c1 = (first formula for $c$ ), c2 = (second formula), etc., replacing $a$ and $b$ with the Desmos variables you set. Desmos will show you numerical values for each $c$ based on your chosen $a$ and $b$ .

Check which $c$ values satisfy the original equation

For each candidate $c$ value (say c1, c2, etc.), type an expression for the right-hand side of the original equation, such as rhs1 = b/(c1^2 - 4), and compare it to your chosen $a$ . The correct formula for $c$ will be the one whose corresponding rhs equals $a$ exactly for your test values (and will continue to work for other positive choices of $a$ and $b$ ).

Step-by-step Explanation

Clear the fraction

Start with the given equation:

a = \frac{b}{c^2 - 4}.

Multiply both sides by $c^2 - 4$ to eliminate the denominator:

a(c^2 - 4) = b.

Isolate the $c^2$ term

Divide both sides of

a(c^2 - 4) = b

by $a$ (which is positive) to isolate the parentheses:

c^2 - 4 = \frac{b}{a}.

Now add $4$ to both sides to solve for $c^2$ :

c^2 = \frac{b}{a} + 4.

Take the square root

From

c^2 = \frac{b}{a} + 4,

take the square root of both sides:

c = \pm \sqrt{\frac{b}{a} + 4}.

At this point, there are two possible roots, one positive and one negative.

Use the fact that $c$ is positive and match the form

The problem states that $c$ is a positive real number, so we only keep the positive root:

c = \sqrt{\frac{b}{a} + 4}.

This matches the correct answer choice.

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Step-by-step Explanation

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Step-by-step Explanation