Question

Hi. I need help with these questions. See image for question. Answer 14 and 13

Answer 1

Answer:

• 13) 36x² + 9x - 4 = 0

• 14) 21x² + 20x + 100 = 0

Step-by-step explanation:

13.

Given

Quadratic equation

• 4x² - 3x - 4 = 0
• With the roots of α and β

To Find

• The quadratic equation with roots of 1/(3α) and 1/(3β)

Solution

The sum and the product of the roots of the given equation:

• α + β = -b/a ⇒ α + β = -(-3)/4 = 3/4
• αβ = c/a ⇒ αβ = -4/4 = - 1

New equation is:

• (x - 1/(3α))(x - 1/(3β)) = 0
• x² - (1/(3α) + 1/(3αβ))x + 1/(3α3β) = 0
• x² - ((3α + 3 β)/(3α3β))x + 1/(3α3β) = 0
• x² - ((α + β)/(3αβ))x + 1/(9αβ) = 0
• x² - (3/4)/(3(-1))x + 1/(9(-1)) = 0
• x² + 1/4x - 1/9 = 0
• 36x² + 9x - 4 = 0

===================

14.

Given

Quadratic equation

• 3x² +2x + 7 = 0
• With the roots of α and β

To Find

• The quadratic equation with roots of α + 1/β and β + 1/α

Solution

The sum and the product of the roots of the given equation:

• α + β = -b/a ⇒ α + β = -2/3
• αβ = c/a ⇒ αβ = 7/3

New equation is:

• (x - (α + 1/β))(x - (β + 1/α)) = 0
• x² - (α + 1/β + β + 1/α)x + (α + 1/β) (β + 1/α) = 0
• x² - (α + β + (α + β)/αβ )x + αβ + 1/αβ + 2 = 0
• x² - (-2/3 - (2/3)/(7/3))x + 7/3 + 1/(7/3) + 2 = 0
• x² - (-2/3 - 2/7)x + 7/3 + 3/7 + 2 = 0
• x² + (14 + 6)/21x + (49 + 9 + 42/21) = 0
• x² + 20/21x + 100/21 = 0
• 21x² + 20x + 100 = 0