Question

A) The quadratic 3x² - 12x - 11 can be written in the form 3(x + a)² + b

Find the values of a and b.
b) Given that the solutions of the equation 3x² - 12x-11 = 0
can be written as c± √d, where c and d are rational numbers.
Find the values of c and d.
Note: Please write your answer for d as a fraction in its simplest form.

Answer 1

see explanation

Explanation:

(a)

3x² - 12x - 11 ( factor out 3 from the first 2 terms )

= 3(x² - 4x) - 11

using the method of completing the square

add/subtract ( half the coefficient of the x- term )² to x² - 4x

= 3(x² + 2(- 2)x + 4 - 4) - 11

= 3(x - 2)² - 12 - 11

= 3(x - 2)² - 23 ← in the form 3(x + a)² + b

with a = - 2 and b = - 23

(b)

3x² - 12x - 11 = 0 , may be expressed as

3(x - 2)² - 23 = 0 ( add 23 to both sides )

3(x - 2)² = 23 ( divide both sides by 3 )

(x - 2)² =
`(23)/(3)` ( take square root of both sides )

x - 2 = ±
`\sqrt{(23)/(3) }` ( add 2 to both sides )

x = 2 ±
`\sqrt{(23)/(3) }` ← in the form c ±
`√(d)`

with c = 2 and d =
`(23)/(3)`