Question

A square painting has an area of 81x^2 - 90x + 25. A second square painting has an area of 25x^2 + 30x + 9. Part A: What is an expression that represents the difference of the areas of the paintings? Show two different ways to find the solution.Part B: The paintings have areas of 1,600in^2 and 484in^2, respectively. A potential buyer went to an exhibition and argued that the value of xfor both paintings is the same. Is he right? Prove it.

Answer 1

56x² - 120x + 16 ; (9x - 5)² - (5x - 3)²

x in both equations is 5

Explanation:

Painting 1 : 81x^2 - 90x + 25

Painting 2 : 25x^2 + 30x + 9

Expression for the difference between the two paintings :

Painting 1 - painting 2

(81x^2 - 90x + 25) - (25x^2 + 30x + 9)

81x^2 - 90x + 25 - 25x^2 - 30x - 9

81x² - 25x² - 90x - 30x + 25 - 9

56x² - 120x + 16

2.)

81x^2 - 90x + 25 = 1600in²

25x^2 + 30x + 9 = 484 in²

Factorizing each equation :

81x^2 - 90x + 25 = 1600 in²

(9x - 5)^2 = 1600 in²

25x^2 + 30x + 9 = 484 in²

(5x - 3)^2 = 484

(9x - 5)² - (5x - 3)²

2.)

(9x - 5)^2 = 1600

Take square root of both sides

9x - 5 = 40

9x = 40 + 5

9x = 45

x = 45/9

x = 5

(5x - 3)^2 = 484

Take square root of both sides

5x - 3 = 22

5x = 22 + 3

5x = 25

x = 25/5

x = 5