Find the zeros of the following quadratic polynomials and verify the basic relationships between the zeros and the coefficients.

x² + 2x – 143

Let f(x) = x² + 2x – 143

To find out zeros of the given polynomial.

We put f(x) = 0

⇒ x² + 2x – 143 = 0

To find out roots of this polynomial we use splitting of middle term method.

According to this method we need to find two numbers whose sum is 2 and product is – 143.

∴ x² + (13 – 11)x – 143 = 0

⇒x² + 13x – 11x – 143 = 0

⇒ x (x + 13) – 11(x + 13) = 0

⇒ (x – 11) (x + 13) = 0

∴ When, (x – 11) = 0

∴ Then, x = 11.

Again, When, (x + 13) = 0

∴ Then, x = – 13

⇒ Our zeros are α = 11 and β = – 13.

⇒ sum of zeros = α + β = 11 + ( – 13)

⇒ sum of zeros = α + β = – 2

⇒ Product of zeros = αβ = 11 × ( – 13) = – 143

Now, Comparing f(x) = x² + 2x – 143 with standard equation ax² + bx + c.

We get, a = 1, b = 2 and c = – 143.

We can verify,

⇒ Sum of zeros =

i.e. α + β =

∴ α + β = – 2

⇒ Product of zeros =

αβ = – 143

Hence, relationship between zeros and coefficient is verified.