Question

the sum of the roots of a quadratic
equation is 23 and the product of its
roots is 120. find the sum of the square
of its roots​

Answer 1

Concept:

( a + b )² = a² + b² + 2 a b

Given:

We are given that:

Sum of roots = 23

Product of roots = 120

Find:

We need to find the sum of the square of its roots​ i.e. x² + y².

Solution:

Let the first root be x and the second root be y.

ATQ:

x + y = 23

x y = 120

We know that:

( a + b )² = a² + b² + 2 a b

( x + y )² = x² + y² + 2 x y

(23)² = x² + y² + 2(120)

529 = x² + y² + 240

x² + y² = 529 - 240

x² + y² = 289

Therefore, we get that the sum of the square of its roots​ i.e. x² + y² is 289.

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Answer 2

Answer:

The sum of the square of its roots​ is 289.

Step-by-step explanation:

Given that

Roots of  quadratic equation (ax² + by² + c = 0)

• Sum of roots = 23
• Product of roots = 120

To find

• The sum of the square of its roots​

Explanation

According the Question

We have,

• A quadratic equation (ax² + by² + c = 0)

Let's assume the roots of that quadratic equation are α and β.

So,

• Sum of roots (α + β) = 23        ------------------(1)
• Product of roots (α × β) = 120        ------------------(2)

We have to find

• the sum of the square of its roots​, (α²+β²)

We know that,

        (a² + b)² = a² + b² + 2ab

So,

      a² + b² = (a + b)² - 2ab

According this, we can say that    

      α² + β² = (α + β)² - 2αβ          ------------------(3)

Now, putting the values of equation (1) and (2) in equation (3)

So, we get

      α² + β² = (α + β)² - 2αβ          ------------------(3)

      α² + β² = (23)² - 2×120        

      α² + β² = 529 - 240

      α² + β² = 289

So, the sum of the square of its roots​ is 289.

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