Find Rectangle Width: Area (m²+4m-12) with Length (m+6)

Q: Why can't I just guess and check the multiple choice answers?

You can verify your final answer, but guessing doesn't help you learn the method! Plus, you need to understand why $ m-2 $ works by seeing that $ (m+6)(m-2) = m^2+4m-12 $.

Q: How do I know which two numbers multiply to -12 and add to +4?

List factor pairs of -12: (1,-12), (-1,12), (2,-6), (-2,6), (3,-4), (-3,4). Check which pair adds to +4. Here, 6 + (-2) = 4, so we get $ (m+6)(m-2) $.

Q: What if the area expression doesn't factor nicely?

Then you'd need polynomial long division or synthetic division. But in this problem, the quadratic does factor perfectly, which is the hint that factoring is the right approach!

Q: Can I use the quadratic formula instead of factoring?

The quadratic formula finds roots (where the expression equals zero), not the factored form we need here. Factoring directly gives us $ (m+6)(m-2) $ which makes the division easy.

Q: Why does the (m+6) cancel out perfectly?

Because $ m^2+4m-12 $ was specifically designed to have $ (m+6) $ as a factor! When we factor it as $ (m+6)(m-2) $, the $ (m+6) $ terms cancel: $ \frac{(m+6)(m-2)}{m+6} = m-2 $.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Express the adjacent side using M

00:03 We'll use the formula for calculating rectangle area (side times side)

00:08 We'll substitute appropriate values according to the given data and solve for side W

00:16 We'll factor using trinomial, noting the coefficients

00:21 We want to find 2 numbers whose sum equals B (4)

00:26 and their product equals C (-12)

00:32 These are the appropriate numbers, let's put them in parentheses

00:48 Let's reduce what we can

00:52 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

A rectangle has an area equal to

$m^2+4m-12$ cm² and a length of $m+6$ cm.

Determine the length of the width of the rectangle:

2

Step-by-step solution

Observe the rectangle $ABCD$ :

(Drawing - marking the given data regarding AB on it)

Continue and write down the data regarding the rectangle's area and the given side length in mathematical form:

$\begin{cases} \textcolor{red}{S_{ABCD}}= m^2+4m-12 \\ \textcolor{blue}{AB}=m+6 \\ \end{cases}$

(we'll use colors for greater clarity )

Remember that the area of a rectangle whose side lengths (adjacent) are:

$a,\hspace{2pt}b$ is:

$S_{\boxed{\hspace{6pt}}}=a\cdot b$

Therefore the area of the rectangle in the problem (according to the drawing we established at the beginning of the solution) is:

$S_{ABCD}=AB\cdot AD$

Now we are able to insert the previously mentioned data into this expression for area to obtain the equation (for understanding - use the marked colors and the data mentioned earlier accordingly):

$\textcolor{red}{ S_{ABCD}}=\textcolor{blue}{AB}\cdot AD \\ \downarrow\\ \boxed{ \textcolor{red}{ m^2+4m-12}=\textcolor{blue}{(m+6 )}\cdot AD}$

Let's pause for a moment to determine our goal:

Our goal is of course to obtain the algebraic expression for the side adjacent to the given side in the rectangle (denoted by m), meaning we want to obtain an expression for the length of side $AD$ ,

Let's return then to the equation that we previously obtained and proceed to isolate $AD$ . This can be achieved by dividing both sides of the equation by the algebraic expression that is the coefficient of $AD$ , that is by: $(m+6 )$ :

$\boxed{ \textcolor{red}{\textcolor{blue}{(m-2)}\cdot AD= m^2+4m-12}} \hspace{4pt}\text{/:}(m+6 )\\ \downarrow\\ AD=\frac{m^2+4m-12}{m+6 }$

Let's continue to simplify the algebraic fraction that we obtained. We can do this easily by factoring the numerator of the fraction:

$m^2+4m-12$

Apply quick trinomial factoring as shown below:

$m^2+4m-12\leftrightarrow\begin{cases} \boxed{?}\cdot\boxed{?}=-12\\ \boxed{?}+\boxed{?}=4\ \end{cases}\\ \downarrow\\ (m+6)(m-2)$

Therefore (returning to the expression for $AD$ ):

$AD=\frac{m^2+4m-12}{m+6 } \\ \downarrow\\ AD=\frac{(m+6)(m-2)}{m+6 }\\ \downarrow\\ \boxed{AD=m-2}$ (length units)

In the final stage, after we factored the numerator of the fraction and reduced the fraction,

(Drawing - with the found AD length)

Therefore the correct answer is answer D.

3

Final Answer

$m-2$

Key Points to Remember

Essential concepts to master this topic

Area Formula: Rectangle area equals length times width always
Technique: Factor $m^2+4m-12 = (m+6)(m-2)$ to simplify division
Check: Multiply width $(m-2)$ by length $(m+6)$ equals given area ✓

Common Mistakes

Avoid these frequent errors

Attempting polynomial long division without factoring first
Don't jump straight into polynomial long division of $\frac{m^2+4m-12}{m+6}$ = complicated work and errors! This makes simple problems unnecessarily difficult. Always try factoring the numerator first to see if terms cancel.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2-3x-18=0 $$

FAQ

Everything you need to know about this question

Why can't I just guess and check the multiple choice answers?

+

You can verify your final answer, but guessing doesn't help you learn the method! Plus, you need to understand why $m-2$ works by seeing that $(m+6)(m-2) = m^2+4m-12$ .

How do I know which two numbers multiply to -12 and add to +4?

+

List factor pairs of -12: (1,-12), (-1,12), (2,-6), (-2,6), (3,-4), (-3,4). Check which pair adds to +4. Here, 6 + (-2) = 4, so we get $(m+6)(m-2)$ .

What if the area expression doesn't factor nicely?

+

Then you'd need polynomial long division or synthetic division. But in this problem, the quadratic does factor perfectly, which is the hint that factoring is the right approach!

Can I use the quadratic formula instead of factoring?

+

The quadratic formula finds roots (where the expression equals zero), not the factored form we need here. Factoring directly gives us $(m+6)(m-2)$ which makes the division easy.

Why does the (m+6) cancel out perfectly?

+

Because $m^2+4m-12$ was specifically designed to have $(m+6)$ as a factor! When we factor it as $(m+6)(m-2)$ , the $(m+6)$ terms cancel: $\frac{(m+6)(m-2)}{m+6} = m-2$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Factorization questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

Find Rectangle Width: Area (m²+4m-12) with Length (m+6)

Polynomial Division with Factoring Methods

❤️ Continue Your Math Journey!