If (2x + 1)(x – 5) = 2 (x^2 – 1), what is the value of x

If $(2x+1)(x-5)=2(x^2-1)$, what is the value of $x$ ?

  1. $-1/3$
  2. $-1/5$
  3. $0$
  4. $1/3$
  5. $1/2$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 10 of the second Quantitative section of Practice Test 1. Those questions testing our knowledge of Operations with Algebraic Expressions can be kind of tricky, but never fear, PrepScholar has got your back!

Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

Well, looks like we’re given a long equation with $x$’s in it, and we want to solve for $x$. So looks like this question tests how well we can perform Operations with Algebraic Expressions. Let’s keep what we’ve learned about this skill at the tip of our minds as we approach this question.

What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

 

  1. We have a big equation containing $x$’s
  2. We want to solve for the value of $x$

 

Develop a Plan

To solve for $x$, we need to eventually get rid of all of the parentheses around it, and isolate it so that it is alone on one side of the equation. Here, we’ll need to keep in mind the Order of Operations, PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) while we simplify both sides of the equation.

Solve the Question

Let’s start by using the FOIL method (First, Outer, Inner, Last) two expand the left side of the equation to get rid of the two sets of parentheses. Basically, on the left side of the equation we’ll just multiply each term in the first set of parentheses with each term in the second set of parentheses, then add everything together. The four terms are: $2x, 1, x, \and -5$.

$(2x+1)(x-5)$ $=$ $2(x^2-1)$
$ $ $ $
$2x·x+2x·(-5)+1·x+1·(-5)$ $=$ $2(x^2-1)$
$ $ $ $
$2x^2-10x+x-5$ $=$ $2(x^2-1)$
$ $ $ $
$2x^2-9x-5$ $=$ $2x^2-2$

In the last step, we also distributed the $2$ inside the parentheses on the right side. Let’s continue simplifying. We see that both sides have a $2x^2$ term, so let’s subtract $2x^2$ from each side:

$2x^2-9x-5$ $=$ $2x^2-2$
$ $ $ $
$-9x-5$ $=$ $-2$

Now let’s finish isolating the $x$. We’ll add $5$ to both sides of the equation, then divide by $-9$.

$-9x-5$ $=$ $-2$
$ $ $ $
$-9x-5+5$ $=$ $-2+5$
$ $ $ $
$-9x$ $=$ $3$
$ $ $ $
$x$ $=$ $3/{-9}$
$ $ $ $
$x$ $=$ ${1·3}/{-3·3}$
$ $ $ $
$x$ $=$ $-1/3$

The correct answer is A, $-1/3$.

What Did We Learn

Slow and steady wins the race. Though there were a lot of steps involved in simplifying the equation in this question, none of the steps were too large or complicated. The key here is to move quickly between steps so that we can finish this question in a reasonable amount of time.

One way to check to see how efficiently you’re solving a question like this is to watch your pen or pencil. If it stays up from the paper for more than a couple seconds while working through the algebraic steps, that’s an inefficiency that we should try to improve before test day. Practice makes perfect!

 

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