Question

Find the quadratic equation the sum of whose roots is 1 and the sum of the squares of the roots is 13.

Answer 1

Answer :-

x² - x - 6

Step-by-step explanation :-

Let α and β be the 2 roots

In the given question,

• α + β = 1
• α² + β² = 13

We know that (a + b)² = a² + b² + 2ab

Apply this algebraic identity here.

  (α + β)² = α² + β² + 2(αβ)

⇒ (1)² = 13 + 2(αβ)

⇒ 1 - 13 = 2(αβ)

⇒ 2(αβ) = -12

⇒ αβ = -6

∴ Product of roots = -6

Quadratic equation = x² - (α + β)x + αβ

⇒ Quadratic Equation = x² - (1)x + (-6)

⇒ Quadratic Equation = x² - x - 6

∴ Quadratic Equation = x² - x - 6

Answer 2

EXPLANATION:-

Let the roots be  α and β

Here in this question:-

α + β = 1

α² + β² = 13

Using the identity:(a + b)² = a² + b² + 2ab

Let's apply this identity here:-

(α + β)² = α² + β² + 2(αβ)

==> (1)² = 13 + 2(αβ)

==>1 - 13 = 2(αβ)

==> 2(αβ) = -12

==> αβ = -6

So we get that the product of roots is -6

Quadratic equation=x² - (α + β)x + αβ

==>x² - (1)x + (-6)          

==> x² - x - 6

So the quadratic equation is x² - x - 6