Activity 2
1. Logan was given a rectangle with length $(x-3)$ and width $(2x+5)$ and asked to find the area.
Logan's final answer for the area of the rectangle: $2x^2-30x-15$. Logan made a mistake. Explain his mistake and then provide the correct answer. Be specific about what he did and what he should have done instead in your explanation.
2. Logan realized what he did wrong and is ready to try factoring. He is given a rectangle with the area of $3x^2-13x-10$ and asked to find the dimensions. What should Logan get for the length and width (factors) of the rectangle? Show your work.
Answer (1)
Logan incorrectly calculated the area of the rectangle with sides ( x − 3 ) and ( 2 x + 5 ) . The correct area is found by expanding the product: ( x − 3 ) ( 2 x + 5 ) = 2 x 2 − x − 15 .
Logan's mistake was in combining the x terms; he incorrectly calculated 5 x − 6 x as − 30 x .
To find the dimensions of the rectangle with area 3 x 2 − 13 x − 10 , factor the quadratic expression.
The factored form of 3 x 2 − 13 x − 10 is ( 3 x + 2 ) ( x − 5 ) , so the dimensions of the rectangle are ( 3 x + 2 ) and ( x − 5 ) .
( 3 x + 2 ) and ( x − 5 )
Explanation
Analyzing the Problem Let's analyze Logan's mistake in finding the area of the rectangle. The length is ( x − 3 ) and the width is ( 2 x + 5 ) . To find the area, we need to multiply these two expressions. Logan's answer was 2 x 2 − 30 x − 15 , which is incorrect. We will expand the product ( x − 3 ) ( 2 x + 5 ) to find the correct area and then pinpoint Logan's error.
Calculating the Correct Area To find the area of the rectangle, we multiply the length and width: ( x − 3 ) ( 2 x + 5 ) = x ( 2 x ) + x ( 5 ) − 3 ( 2 x ) − 3 ( 5 ) = 2 x 2 + 5 x − 6 x − 15 = 2 x 2 − x − 15 The correct area of the rectangle is 2 x 2 − x − 15 .
Identifying Logan's Mistake Comparing the correct area ( 2 x 2 − x − 15 ) with Logan's answer ( 2 x 2 − 30 x − 15 ), we can see that Logan made a mistake in calculating the middle term. It seems he might have incorrectly multiplied or combined the terms 5 x and − 6 x . The correct combination is 5 x − 6 x = − x , but Logan somehow got − 30 x .
Factoring the Quadratic Expression Now, let's factor the quadratic expression 3 x 2 − 13 x − 10 to find the dimensions of another rectangle. We are looking for two binomials ( a x + b ) and ( c x + d ) such that when multiplied, they give 3 x 2 − 13 x − 10 .
We need to find a , b , c , and d such that: a c = 3 a d + b c = − 13 b d = − 10 Since 3 is a prime number, a and c must be 3 and 1 (or 1 and 3). Let's try a = 3 and c = 1 . Now we have ( 3 x + b ) ( x + d ) .
We need to find b and d such that: 3 d + b = − 13 b d = − 10 Possible pairs for b and d are: (1, -10), (-1, 10), (2, -5), (-2, 5) Let's test the pair (2, -5): 3 ( − 5 ) + 2 = − 15 + 2 = − 13 This works! So, b = 2 and d = − 5 . Therefore, the factors are ( 3 x + 2 ) ( x − 5 ) .
Finding the Dimensions of the Rectangle The dimensions (length and width) of the rectangle are ( 3 x + 2 ) and ( x − 5 ) .
Final Answer In summary, Logan's mistake was in calculating the middle term when expanding ( x − 3 ) ( 2 x + 5 ) . The correct area is 2 x 2 − x − 15 . The factors of 3 x 2 − 13 x − 10 are ( 3 x + 2 ) and ( x − 5 ) , which represent the length and width of the rectangle.
Examples
Understanding how to calculate the area of a rectangle and factor quadratic expressions is very useful in real life. For example, if you're planning to build a rectangular garden, you need to calculate the area to determine how much soil to buy. Also, if you know the area of the garden and want to find possible dimensions, you would need to factor the quadratic expression representing the area. This is also applicable in construction, interior design, and many other fields where area and dimensions are important.
