Which expression is equivalent to where ?

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SAT Equivalent Expressions Question 47 | Free SAT Prep | Aniko

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Question 47·Medium·Equivalent Expressions

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Which expression is equivalent to

\frac{12x^2+5x-3}{4x+3},

where $x\neq-\dfrac34$ ?

When simplifying a rational expression where the numerator is a quadratic and the denominator is a linear binomial, first try to factor the numerator so that the denominator appears as a factor. Set up $(\text{denominator})(ax+b)$ , expand, and match coefficients to solve for $a$ and $b$ . Once you see the numerator as (denominator) × (other factor), you can cancel the common binomial (using any given domain restriction to justify the cancellation) and read off the remaining factor as your simplified expression, checking against the answer choices.

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Think about factoring

Instead of doing long division, ask: can the quadratic numerator $12x^2+5x-3$ be written as a product of two binomials?

Use the denominator as a clue

Since the denominator is $4x+3$ , consider whether $4x+3$ could be one of the factors of the numerator. What would the other factor have to look like?

Match coefficients

If you write $(4x+3)(ax+b)$ and expand, you get $4ax^2 + (4b+3a)x + 3b$ . Match these coefficients to $12x^2+5x-3$ to solve for $a$ and $b$ .

After factoring, simplify

Once the numerator is written as $(4x+3)(\text{something})$ , what happens to the $4x+3$ factor in the numerator and denominator, given that $x\neq -\tfrac{3}{4}$ ?

Desmos Guide

Enter the original expression

In Desmos, type f(x) = (12x^2 + 5x - 3) / (4x + 3) to graph the given rational expression. Note that there will be a hole or vertical asymptote at $x = -0.75$ due to the denominator.

Compare each answer choice

For each option, enter it as a separate function, for example g(x) = 3x - 1, h(x) = (3x - 1)/4, etc. Compare the graphs: the correct choice will have a graph that exactly overlaps the graph of f(x) everywhere it is defined (except at $x = -0.75$ , where f(x) is undefined).

Alternative: check algebraically with Desmos

You can also type expressions like (12x^2 + 5x - 3) / (4x + 3) - (3x - 1) for each candidate. The correct answer will make this difference equal to 0 for all values of $x$ where the original expression is defined (you can confirm by looking at the table or graph of this difference).

Step-by-step Explanation

Recognize the goal: simplify a rational expression

You are given the expression

\frac{12x^2+5x-3}{4x+3}

and asked for an equivalent, simpler expression. This means we want to rewrite it, usually by factoring and canceling, not by plugging in numbers.

Try to factor the numerator using the denominator

Notice the denominator is $4x+3$ . A common SAT pattern is that the numerator factors to include the denominator as a factor.

Assume the numerator factors as $(4x+3)(ax+b)$ . Expand this product:

(4x+3)(ax+b) = 4ax^2 + (4b+3a)x + 3b.

We want this to match $12x^2 + 5x - 3$ . So match coefficients:

$4a = 12$ so $a = 3$ .
$3b = -3$ so $b = -1$ .

Now check the middle term with $a=3$ and $b=-1$ :

4b + 3a = 4(-1) + 3(3) = -4 + 9 = 5,

which matches the $5x$ term.

Rewrite the fraction using the factorization

From the previous step, we have

12x^2 + 5x - 3 = (4x+3)(3x-1).

So the original expression becomes

\frac{12x^2+5x-3}{4x+3} = \frac{(4x+3)(3x-1)}{4x+3}.

Now you can see a common factor in the numerator and denominator.

Cancel the common factor (using the domain restriction)

Because $x \neq -\tfrac{3}{4}$ , the denominator $4x+3$ is never zero, so it is safe to cancel the common factor $4x+3$ :

\frac{(4x+3)(3x-1)}{4x+3} = 3x-1.

Thus, the expression is equivalent to $3x-1$ for all allowed values of $x$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation