Q1 of 227 Page 82

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

x2 – 2x – 8

Let f(x) = x2 – 2x – 8

To find out zeros of the given polynomial.


We put f(x) = 0


⇒ x2 – 2x – 8 = 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 8.


∴ x2 – (4 – 2)x – 8 = 0


⇒ x2 – 4x + 2x – 8 = 0


⇒ x(x – 4) + 2(x – 4) = 0


⇒ (x + 2)(x – 4) = 0


∴ x = – 2 and x = 4.


⇒ Our zeros are α = – 2 and β = 4.


⇒ sum of zeros = α + β = – 2 + 4 = 2.


⇒ Product of zeros = αβ = ( – 2) × 4 = – 8.


⇒ Comparing f(x) = x2 – 2x – 8 with standard equation ax2 + bx + c = 0.


We get, a = 1, b = – 2 and c = – 8


We can verify,


⇒ Sum of zeros =


i.e. α + β =


∴ α + β = 2


⇒ Product of zeros =


αβ =


αβ = – 8.


Hence, relationship between zeros and coefficient is verified.


More from this chapter

All 227 →
2

Formulate the following problems as a pair of equations, and hence find their solutions:

In a rectangle, if the length is increased and the breadth is reduced each by 2 cm then the area is reduced by 28 cm2. If the length is reduced by 1 cm and the breadth increased by 2 cm , then the area increases by 33 cm2. Find the area of the rectangle.

 2

Formulate the following problems as a pair of equations, and hence find their solutions:

A train travelled a certain distance at a uniform speed. If the train had been 6 km/hr faster, it would have taken 4 hours less than the scheduled time. If the train were slower by 6 km/hr, then it would have taken 6 hours more than the scheduled time. Find the distance covered by the train.

 1

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

4x2 – 4x + 1

 1

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

6x2 – 3 – 7x