What is the solution of the equation (x – 5)2 + 3(x – 5) + 9 = 0? use u substitution and the quadratic formula to solve.

x = 1/5 (25 + 3 i sqrt(5)) or x = (-3 i sqrt(5) + 25)/5

Step-by-step explanation using the quadratic formula:

Solve for x:

5 (x - 5)^2 + 9 = 0

Expand out terms of the left hand side:

5 x^2 - 50 x + 134 = 0

x = (50 ± sqrt((-50)^2 - 4×5×134))/(2×5) = (50 ± sqrt(2500 - 2680))/10 = (50 ± sqrt(-180))/10:

x = (50 + sqrt(-180))/10 or x = (50 - sqrt(-180))/10

sqrt(-180) = sqrt(-1) sqrt(180) = i sqrt(180):

x = (50 + i sqrt(180))/10 or x = (50 - i sqrt(180))/10

sqrt(180) = sqrt(4×9×5) = sqrt(2^2×3^2×5) = 2×3sqrt(5) = 6 sqrt(5):

x = (i×6 sqrt(5) + 50)/10 or x = (-i×6 sqrt(5) + 50)/10

Factor 2 from 50 + 6 i sqrt(5) giving 2 (3 i sqrt(5) + 25):

x = 1/102 (3 i sqrt(5) + 25) or x = (-6 i sqrt(5) + 50)/10

(2 (3 i sqrt(5) + 25))/10 = (2 (3 i sqrt(5) + 25))/(2×5) = (3 i sqrt(5) + 25)/5:

x = (3 i sqrt(5) + 25)/5 or x = (-6 i sqrt(5) + 50)/10

Factor 2 from 50 - 6 i sqrt(5) giving 2 (-3 i sqrt(5) + 25):

x = 1/5 (25 + 3 i sqrt(5)) or x = 1/102 (-3 i sqrt(5) + 25)

(2 (-3 i sqrt(5) + 25))/10 = (2 (-3 i sqrt(5) + 25))/(2×5) = (-3 i sqrt(5) + 25)/5:

x = 1/5 (25 + 3 i sqrt(5)) or x = (-3 i sqrt(5) + 25)/5

Step-by-step explanation:

To solve :

On putting x-5 = u in the given equation, we get

We will sove this equation using quadratic formula:

For equation of form , roots are given by

On comparing equation with equation , we get

So, roots are

On putting u = x-5 in , we get

let x be the money she spent so that she can earn the free t-shirt.

since, she already spent $112.49

⇒

also, as per the statement: jennifer must spend $250 or more in order to qualify for a free t-shirt at the store.

then,

we have an inequality i.,e

subtract 112.49 from both sides we get;

therefore, $137. 51 more money does jennifer need to spend so that she can earn the free t-shirt