Roots And Coefficients | MCQs

Explanation

Given quadratic equation: 9x² - 27x + k = 0

Step 1: Find the sum and product of roots

Sum of roots (α + β) = -(-27)/9 = 3

Product of roots (αβ) = k/9

Step 2: Use the given condition

2α + 5β = 7

We also know α + β = 3

Step 3: Express α in terms of β or vice versa

From α + β = 3, α = 3 - β

Step 4: Substitute α in the given condition

2(3 - β) + 5β = 7

6 - 2β + 5β = 7

3β = 1

β = 1/3

Step 5: Find α

α = 3 - β = 3 - 1/3 = 8/3

Step 6: Find k using the product of roots

αβ = k/9

(8/3)(1/3) = k/9

8/9 = k/9

k = 8

Explanation

Given the quadratic equation mx² - 5x + n = 0:

Sum of roots = -(-5/m) = 5/m

Product of roots = n/m

Given both sum and product of roots are equal to 10:

5/m = 10 => m = 1/2

n/m = 10 => n = 10m

n = 10 * (1/2)

n = 5

Explanation

One root = 2  

Equation: 2x² + ax + 6 = 0

Using factor theorem, substitute x = 2:

2(2)² + a(2) + 6 = 0  

⇒ 8 + 2a + 6 = 0  

⇒ 2a + 14 = 0  

⇒ 2a = -14  

⇒ a = -7