  # Find the Quadratic Polynomial Whose Zeros are 3+√5 and 3-√5

By BYJU'S Exam Prep

Updated on: October 17th, 2023 x^2 – 6x + 4 has Zeros as 3+√5 and 3-√5.

To find the result, we will start with the general form of a quadratic polynomial. Thereafter, we will substitute the given zeros into the equation. Then, we will simplify the expression inside the brackets and use the difference of squares identity to expand the brackets.
Finally, we will simplify the constants to get the desired result.

Table of content ## Find the Quadratic Polynomial Whose Zeros are 3+√5 and 3-√5

Solution:

To find the quadratic polynomial with the given zeros, we can use the fact that if α and β are the zeros of a quadratic polynomial, then the polynomial can be expressed as:

P(x) = (x – α)(x – β)

In this case, the zeros are 3 + √5 and 3 – √5. Let’s substitute these values into the equation:

P(x) = (x – (3 + √5))(x – (3 – √5))

Simplifying the expression inside the brackets:

P(x) = (x – 3 – √5)(x – 3 + √5)

Now, let’s apply the difference of squares:

P(x) = [(x – 3) – √5][(x – 3) + √5]

Using the identity (a – b)(a + b) = a^2 – b^2:

P(x) = (x – 3)^2 – (√5)^2

Simplifying further:

P(x) = (x – 3)^2 – 5

Expanding the square:

P(x) = x^2 – 6x + 9 – 5

Simplifying the constants:

P(x) = x^2 – 6x + 4

Therefore, the quadratic polynomial with zeros 3 + √5 and 3 – √5 is P(x) = x^2 – 6x + 4. GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com