Question

• Class - 10 •

Find the quadratic polynomials whose zeroes are
3 ± √2

Answer 1

Answer: p(x) = x² - 6x + 7

Given the zeroes of a quadratic polynomials as:

1. 3 + √2
2. 3 - √2

We know, when the coefficient of x² is 1 then the coefficient of x is negative of the sum of the zeroes while the constant term is the product of the zeroes. Which is given by:

• p(x) = x² - (Sum of zeroes)x + Product of zeroes

Now, Let's find the sum and product of the zeroes,

⇒ Sum of zeroes = (3 + √2) + (3 - √2)

⇒ Sum of zeroes = 3 + √2 + 3 - √2

⇒ Sum of zeroes = 3 + 3

⇒ Sum of zeroes = 6

Further, Let's find the product of zeroes,

⇒ Product of zeroes = (3 + √2)(3 - √2)

⇒ Product of zeroes = (3)² - (√2)²

As,

• (a + b)(a - b) = a² - b²

⇒ Product of zeroes = 9 - 2

⇒ Product of zeroes = 7

Now that we have got the sum and the product of the zeroes let's substitute them in the formula discussed earlier.

⇒ p(x) = x² - 6x + 7 = 0

Hence, It is the required polynomial.

Some Information :-

1. We can find the zeroes of a quadratic polynomial using the quadratic formula where a, b and c are given by

• a = Coefficient of x²
• b = Coefficient of x
• c = constant term

Answer 2

Step-by-step explanation:

Answer: p(x) = x² - 6x + 7

Given the zeroes of a quadratic polynomials as:

3 + √2

3 - √2

We know, when the coefficient of x² is 1 then the coefficient of x is negative of the sum of the zeroes while the constant term is the product of the zeroes. Which is given by:

p(x) = x² - (Sum of zeroes)x + Product of zeroes

Now, Let's find the sum and product of the zeroes,

⇒ Sum of zeroes = (3 + √2) + (3 - √2)

⇒ Sum of zeroes = 3 + √2 + 3 - √2

⇒ Sum of zeroes = 3 + 3

⇒ Sum of zeroes = 6

Further, Let's find the product of zeroes,

⇒ Product of zeroes = (3 + √2)(3 - √2)

⇒ Product of zeroes = (3)² - (√2)²

As,

(a + b)(a - b) = a² - b²

⇒ Product of zeroes = 9 - 2

⇒ Product of zeroes = 7

Now that we have got the sum and the product of the zeroes let's substitute them in the formula discussed earlier.

⇒ p(x) = x² - 6x + 7 = 0

Hence, It is the required polynomial.

Some Information :-

We can find the zeroes of a quadratic polynomial using the quadratic formula where a, b and c are given by

a = Coefficient of x²

b = Coefficient of x

c = constant term

hope this helps you!!

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