Question

if alpha + beta is equal to 8 and Alpha square plus beta square is equal to 34 find the quadratic equation whose roots are alpha and beta

Answer 1

alpha + beta=8
(alpha+beta)^2=34
Alpha square plus beta square is equal to alpha + beta whole square minus 2 alpha beta
therefore, 34=(8)^2- 2×alpha×beta
therefore, 34 = 64 - 2× alpha× beta
therefore, 2×alpha×beta= 64-34
therefore, 2×Alpha×Beta=30
therfore, alpha×beta=30/2 = 15
The required quadratic equation is x^2 - (alpha + beta)x + alpha×beta=0 answer is x^2 - 8 x + 15=0


^ this means raise to the power

Answer 2

The required quadratic equation is x²-8x+15

• Given α+β = 8

and α²+β² = 34

• we know the formula of (α²+β²) = α²+β²+2αβ

• So, α²+β² can be written as

α²+β² = (α+β)²-2αβ

• So, (α+β)²-2αβ = 34 ...(1)

• We had the value of α+β = 8

•So,from (1)

(8)²-2αβ = 34

64-2αβ = 34

2αβ = 64-34 = 30

• Therefore, αβ = 15

• we know the quadratic equation having roots α and β is

x²-(α+β)x+αβ = 0 ...(2)

• We have got αβ = 15,(α+β) = 8

• Substituting the values in equation (2)

x²-(8)x+15 = 0

•Therefore,the quadratic equation is x²-8x+15 = 0.