Solve for x: Rectangle Area x²-13 with Dimensions (x-4) and (x+1)

Question

Below is a rectangle.

x>0

The area of the rectangle is x213 x^2-13 .

Calculate x.

x-4x-4x-4x+1x+1x+1

Video Solution

Solution Steps

00:00 Find X
00:03 Use the formula for calculating rectangle area (side times side)
00:11 Substitute the area according to the given data and solve for X
00:20 Expand brackets properly, multiply each term by each term
00:31 Simplify what's possible
00:38 Collect like terms
00:42 Isolate X
00:58 And this is the solution to the question

Step-by-Step Solution

First, let's recall the formula for calculating the area of a rectangle with sides of length a,b (length units):

S=ab S_{\boxed{\hspace{8pt}}}=a\cdot b

Therefore, by direct calculation, for the rectangle shown in the drawing with side lengths:

x+1,x4 x+1,\hspace{6pt}x-4 (length units),

The expression for the area is:

S=(x+1)(x4) S_{\boxed{\hspace{8pt}}}=(x+1)(x-4)

However, from the given information, we know that the expression for the area of the rectangle in the drawing is:

S=x213 S_{\boxed{\hspace{8pt}}}=x^2-13

Therefore, we can conclude the existence of the equation:

(x+1)(x4)=x213 (x+1)(x-4)=x^2-13

Now, in order to simplify the equation, let's recall the expanded distribution law:

(a+b)(c+d)=ac+ad+bc+bd (a+b)(c+d)=ac+ad+bc+bd

Let's continue and solve the equation we got. First, we'll open the parentheses on the left side, then we'll move terms and combine like terms, and solve the simple equation that results:

(x+1)(x4)=x213x24x+x4=x2133x=10/:(-3)x=3 (x+1)(x-4)=x^2-13 \\ x^2-4x+x-4=x^2-13\\ -3x=-10\hspace{9pt}\text{/:(-3)}\\ \boxed{x=3} (length units),

Note- this solution for the unknown does not contradict the domain of definition (where the side lengths must be positive, as required) and the area obtained by substituting it into the given expression for the area in the problem:

S=x213S=5213=2513=12 S_{\boxed{\hspace{8pt}}}=x^2-13 \\ \downarrow\\ S_{\boxed{\hspace{8pt}}}=5^2-13 =25-13=\boxed{12} (area units)

Indeed positive, as expected.

Therefore, the correct answer is answer C.

Answer

x=3 x=3