Question

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Answer 1

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:

• Any quadratic equation of the form, ax² + bx + c = 0 can be solved using the formula x = -b ± √b² - 4ac / 2a. Here a, b, and c are the coefficients of the x², x, and the numeric term respectively.

• We have to solve all of the five equations to be able to match the equations with their solutions.

• 2x² - 8x + 5, here a = 2, b = -8, c = 5.                                                  x = -b ± √b² - 4ac / 2a = -(-8) ± √(-8)² - 4(2)(5) / 2(2) = 8 ± √64 - 40/4.     24 can also be written as 4 × 6 and √4 = 2. So                                                                                     x = 8 ± 2√6 / 2×2= 4±√6/2.

• 2x² - 10x + 3, here a = 2, b = -10, c = 3.                                                   x =-b ± √b² - 4ac / 2a =-(-10) ± √(-10)² - 4(2)(3) / 2(4) = 10 ± √100 + 24/4. 124 can also be written as 4 × 31 and √4 = 2. So                                                                              x = 10 ± 2√31 / 2×2 = 5 ± √31 /2.

• 2x² - 8x - 3, here a = 2, b = -8, c = -3.                                                    x = -b ± √b² - 4ac / 2a = -(-8) ± √(-8)² - 4(2)(-3) / 2(2) = 8 ± √64 + 24/4.     88 can also be written as 4 × 22 and √4 = 2. So                                                                             x = 8 ± 2√22 / 2×2 = 4± √22/2.
• 2x² - 9x - 1, here a = 2, b = -9, c = -1.                                                     x = -b ± √b² - 4ac / 2a = -(-9) ± √(-9)² - 4(2)(-1) / 2(2) = 9 ± √81 + 8/4.                                          x = 9 ± √89 / 4.

• 2x² - 9x + 6, here a = 2, b = -9, c = 6.                                                    x = -b ± √b² - 4ac / 2a = -(-9) ± √(-9)² - 4(2)(6) / 2(2) = 9 ± √81 - 48/4.                                                                             x = 9 ± √33 / 4 .

Answer 2

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; https://brainly.com/question/8649555