# Find the Zeros of the Following Quadratic Polynomials x²+7x+10 and Verify the Relationship between the Zeros and the Coefficients

By BYJU'S Exam Prep

Updated on: October 17th, 2023

Find the zeros of the following quadratic polynomials x²+7x+10 and verify the relationship between the zeros and the coefficients.

To solve this problem, we will use the quadratic formula. For this we will have to identify the coefficients, i.e., a = 1, b = 7, c = 10 and put in x = (-b ± √(b² – 4ac)) / (2a)

On simplifying, we will get the two different zeros of the polynomial which can be again verified with the help of Vieta’s formula.

## Find the Zeros of the Following Quadratic Polynomials x²+7x+10 and Verify the Relationship between the Zeros and the Coefficients

Solution:

To find the zeros of the quadratic polynomial x² + 7x + 10, we can use the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a)

Comparing the polynomial to the standard quadratic form ax² + bx + c, we have a = 1, b = 7, and c = 10. Substituting these values into the quadratic formula, we get:

x = (-(7) ± √((7)² – 4(1)(10))) / (2(1))

Simplifying further:

x = (-7 ± √(49 – 40)) / 2

x = (-7 ± √9) / 2

x = (-7 ± 3) / 2 = -2

This gives us two possible solutions:

x1 = (-7 + 3) / 2 = -2

x2 = (-7 – 3) / 2 = -5

Therefore, the zeros of the quadratic polynomial x² + 7x + 10 are x = -2 and x = -5.

Now, let’s verify the relationship between the zeros and the coefficients using Vieta’s formulas:

The sum of the zeros is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term:

Sum of zeros = -(7/1) / (1/1) = -7

The product of the zeros is equal to the constant term divided by the coefficient of the quadratic term:

Product of zeros = (10/1) / (1/1) = 10

Indeed, the sum of the zeros, -7, matches the coefficient of the linear term, and the product of the zeros, 10, matches the constant term. Therefore, the relationship between the zeros and the coefficients is verified.