Question

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

Answer 1

Answer:
`x=(7\pm 3i√(3))/(2)`

Step-by-step explanation:

Since, given quadratic function,
`(x-5)^2 + 3(x-5) + 9 = 0` -----(1)

let us consider,
`x-5=p` -----(2)

Put this value in equation (1),

We get,
`p^2+3 p+9=0`, which is also a quadratic equation.

Since, quadratic formula for finding the root of quadratic equation of type
`ax^2+bx+c=0` is,
`x=\frac{-b\pm√(b^2-4ac)} {2a}`

Here, a=1, b=3 and c=9, so,
`p=\frac{-3\pm√(3^2-4*1 *9)} {2*1}`

Thus,
`p=\frac{-3\pm3√(-3)} {2}` ⇒
`p=\frac{-3\pm3i√(3)} {2}` (because √-1=i)

Now, from equation(2)
`x-5=\frac{-3\pm3i√(3)} {2}`

⇒
`x=\frac{-3\pm3i√(3)} {2}+5=(7\pm 3i√(3))/(2)`

⇒
`x=(7\pm 3i√(3))/(2)`

Answer 2

(x – 5)² + 3(x – 5) + 9 = 0

x² - 10x + 25 + 3x - 15 + 9 = 0

x² - 7x - 16 = 0


`x = (7 + √((-7)^2-4(1)(-16)) )/(2)` or
`x = (7 - √((-7)^2-4(1)(-16)) )/(2)`


`x = (7 + 3i √(3) )/(2)` or
`x = (7 - 3i √(3) )/(2)`

(Answer B)