Question

If alpha and beta are the zeroes of the quadratic polynomial 4x^2 - 5x -1, find the value of : alpha square + beta square

Answer 1

ANSWER:

• Value of the above expression is 33/16

GIVEN:

• P(x) = 4x²-5x-1

TO FIND:

• α²+β²

SOLUTION:

P(x) = 4x²-5x-1

Here:

=> (α+β) = -(Coefficient of x)/Coefficient of x²

=> (α+β) = -(-5)/4

=> (α+β) = 5/4

=> Product of zeroes (αβ) = Constant term/ Coefficient of x²

=> αβ = -1/4

Now :

=> α²+β² = (α+β)²-2αβ

Putting the values:

$\implies \: \alpha {}^{2} + \beta {}^{2} = ( \dfrac{5}{4} ) {}^{2} - 2 \times \dfrac{ - 1}{4} \\ \\ \implies \: \alpha {}^{2} + \beta {}^{2} = \dfrac{25}{16} + \dfrac{1}{2} \\ \\ \implies \: \alpha {}^{2} + \beta {}^{2} = \dfrac{33}{16}$

Value of the above expression is 33/16

NOTE:

Some important formulas:

(a+b)² = a²+b²+2ab

(a-b)² = a²+b²-2ab

(a+b)(a-b) = a²-b²

(a+b)³ = a³+b³+3ab(a+b)

(a-b)³ = a³-b³-3ab(a-b)

a³+b³ = (a+b)(a²+b²-ab)

a³-b³ = (a-b)(a²+b²+ab)

(a+b)² = (a-b)²+4ab

(a-b)² = (a+b)²-4ab