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The quadratic equations and their solutions are;

9 ± √33 /4 = 2x² - 9x + 6.

4 ± √6 /2 = 2x² - 8x + 5.

9 ± √89 /4 = 2x² - 9x - 1.

4 ± √22 /2 = 2x² - 8x - 3.

Explanation:

The right solutions that matches each of the given quadratic equations are;

(9 ± √33)/4  → 2x² - 9x + 6.

(4 ± √6)/2 → 2x² - 8x + 5

(9 ± √89)/4 → 2x² - 9x - 1

(4 ± √22)/2 → 2x² - 8x - 3.

The formula to find the roots of a quadratic equation of the form, ax² + bx + c = 0 is;

x = [-b ± √(b² - 4ac)]/2a

where;

a, b, and c are the coefficients of the x², x, and the numeric term respectively.

Applying the quadratic formula to the given quadratic equations;

1) 2x² - 8x + 5;

x = [-(-8) ± √(-8)² - 4(2)(5)]/(2(2))

x = [8 ± √(64 - 40)]/4

x = [8 ± √(24)]/4

x = (8 ± 2√6)/4

x = (4 ± √6)/2  

2) 2x² - 10x - 3

x = [-(-10) ± √((-10)² - 4(2)(-3))]/2(2)

x = [10 ± √(100 + 24)]/4

x = [10 ± √(124)]/4

x = (10 ± 2√31)/4

x = (5 ± √31)/2

3) 2x² - 8x - 3

x = [-(-8) ± √((-8)² - 4(2)(-3))]/2(2)

x = [8 ± √(64 + 24)]/4  

x = [8 ± √(88)]/4                                                                          

x =  [8 ± 2√(22)]/4  

x = (4 ± √22)/2

4) 2x² - 9x - 1

x = [-(-9) ± √((-9)² - 4(2)(-1))]/2(2)

x = [9 ± √(81 + 8)]/4

x = (9 ± √89)/4

5) 2x² - 9x + 6

x = [-(-9) ± √((-9)² - 4(2)(6))]/2(2)

x = [9 ± √(81 - 48)]/4                                                                          

x = (9 ± √33)/4

Read more about quadratic formula at; link

b: |12 – x| ≤ 0.25

step-by-step explanation:      

let x be the possible lengths in inches, of the ribbons jaime cut.

we have been given that jaime is cutting strips of ribbon for a project. he tries to cut each piece to be 12 inches long, but sometimes he cuts it too long or too short by as much as 0.25 inch.  

we can represent this information in an inequality as:

by the definition of absolute value function we will get,

upon combining our inequalities we will get,

therefore, the inequality   describes all the possible lengths (x), in inches, of the ribbons jaime cut.

step-by-step explanation: