A rectangular painting has an area of 420 square inches. If the length is (2x + 5) inches and the width is (x) inches, then what is the value of x? Use completing-the-square and round the value of x to the nearest tenth. (Hint: You may find 2 solutions, but you only need the positive solution because the length and width must be positive values)

12.1 inches

Step-by-step explanation:

Since the length of the rectangular painting is (2x + 5) inches and its width is x inches, it area A = (2x + 5)x square inches = 2x² + 5x.

Also, A = 420 square inches. So,

2x² + 5x = 420

2x² + 5x - 420 = 0

diving through by 2, we have

2x²/2 + 5x/2 - 420/2 = 0/2

x² + 5x/2 - 210 = 0

adding halt the coefficient of x squared to both sides, that is (5/2 ÷ 2)² = (5/4)² to both sides, we have

x² + 5x/2 + (5/4)² - 210 = 0 + (5/4)²

Simplifying, we have

(x + 5/2)² - 210 = 25/16

adding 210 to both sides, we have

(x + 5/2)² - 210 + 210 = 25/16 + 210

(x + 5/2)² = 25/16 + 210

(x + 5/2)² = (25 + 3360)/16

(x + 5/2)² = 3385/16

taking square root of both sides, we have

√(x + 5/2)² = ±√(3385/16)

(x + 5/2) = ±√3385/4

(x + 5/2) = ±58.18075/4

(x + 5/2) = ±14.55

x = -5/2 ±14.55

x = -2.5 ±14.55

x = -2.5 - 14.55 or -2.5 + 14.55

x = - 17.05 or 12.05

Since x = width and it cannot be negative, we choose the positive answer.  So, x = 12.05 inches ≅ 12.1 inches (to the nearest tenth)