One of the roots of the equation x^2 + kx – 6 = 0 is 3, and k is a

One of the roots of the equation $x^2+kx-6=0$ is $3$, and $k$ is a constant.

 

Quantity A $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ Quantity B
$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$
The value of $k$ $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $-1$
  1. Quantity A is greater.
  2. Quantity B is greater.
  3. The two quantities are equal.
  4. The relationship cannot be determined from the information given

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 8 of Section 4 of Practice Test 1. Those questions testing our Solving Quadratic Equations knowledge 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 your paper.

For questions looking for the root of an equation that has a variable squared $(x^2)$, we should expect the problem to involve our skill at Solving Quadratic Equations. 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 quadratic equation with $x$ and $k$
  2. One root of the equation is $3$
  3. We want to compare $k$ to a certain value

 

Develop a Plan

Let’s start with a top-down approach, where we will begin with what we’re looking for and work down to the details of what we’re given in this question. We want to compare $k$ to the value $-1$. We see $k$ in the equation: $x^2+kx-6=0$. We know that if we had a value of $x$ to plug into this equation, then we could just solve it for $k$. So let’s see if we can find a value for $x$ satisfying the quadratic equation.

We know that the root of a quadratic equation tells us that if we plug in the value of the root for $x$, then the equation will equal $0$. So let’s solve this question by plugging in $x=-1$ and then simplify the equation until we can solve for $k$, then compare $k$ to $-1$.

Solve the Question

$x^2+kx-6$ $=$ $0$
$3^2+k·3-6$ $=$ $0$
$9+3k-6$ $=$ $0$
$3+3k$ $=$ $0$
$3k$ $=$ $-3$
$k$ $=$ $-1$

Since Quantity A ($k$) is $-1$, we can see that both quantities are equal. Sothe correct answer is C, the two quantities are equal.

What Did We Learn

A quadratic equation has a variable squared (in our question, it was $x^2$). The root of a quadratic equation tells us that when we plug in the root for $x$, the equation is equal to $0$.

 

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