VERIFYING THE RELATION BETWEEN ZEROES AND COEFFICIENTS

Verifying the relation between zeroes and coefficients : 

The basic relationship between zeroes and coefficient of quadratic equation ax2 + bx+ c  =  0 are 

Sum of zeroes  =  α + β  =  -b/a 

Product of zeroes  =  α β  =  c/a

Here α  and β are zeroes of the quadratic polynomial. a, b and c are the coefficients of x2, x and constant terms respectively.

Verifying the Relation Between Zeroes and Coefficients - Example problems

Question 1 :

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and coefficients

(i) x² – 2 x – 8

Solution :

First, let us find the zeroes of the polynomial. 

x² – 2 x – 8 = 0

(x – 4) ( x + 2) = 0

x - 4  =  0

x  =  4

x + 2  =  0

x  =  -2

So, α = 4 and β = -2

Now we are going to verify the relationship between these zeroes and coefficients

x² – 2 x – 8 = 0

ax² + b x + c = 0

a = 1   b = -2   c = -8

Sum of zeroes α + β = -b/a

4 + (-2) = -(-2)/1

2 = 2

Product of zeroes α β = c/a

4(-2) = -8/1

-8 = -8

(ii) 4 s² – 4 s + 1 

Solution :

First, let us find the zeroes of the quadratic polynomial.

4 s² – 4 s + 1

(2 s – 1) ( 2 s - 1) = 0

2 s – 1 = 0    

2 s = 1

s = 1/2,   s = 1/2

So,  α = 1/2 and β = 1/2

Now we are going to verify the relationship between these zeroes and coefficients

4 s² – 4 s + 1 = 0

ax² + b x + c = 0

a = 4   b = -4   c = 1

Sum of zeroes α + β = -b/a

(1/2) + (1/2) = -(-4)/4

2/2 = 1

1 = 1

Product of zeroes α β = c/a

(1/2)(1/2) = 1/4

1/4 = 1/4

(iii)  6 x² – 3 – 7 x 

Solution :

First, let us find the zeroes of the quadratic polynomial.

6 x² – 3 – 7 x = 0

6 x²  - 7 x – 3 = 0

3x + 1  =  0

3x  =  -1

x  =  -1/3

2x - 3  =  0

2x  =  3

x  =  3/2

So, α = -1/3 and β = 3/2

Now we are going to verify the relationship between these zeroes and coefficients

6 x²  - 7 x – 3 = 0

ax² + b x + c = 0

a = 6   b = -7   c = -3

Sum of zeroes α + β = -b/a

(-1/3) + (3/2) = -(-7)/6

(- 2 + 9)/6 = 7/6

7/6 = 7/6

Product of zeroes α β = c/a

(-1/3)(3/2) = -3/6

-1/2 = -1/2

(iv)  4 u² + 8 u 

Solution :

Find let us find the zeroes of the quadratic polynomial.

4 u² + 8 u = 0

4 u (u + 2) = 0

 4 u = 0       u + 2 = 0

   u = 0            u = -2

So, α = 0 and β = -2

Now we are going to verify the relationship between these zeroes and coefficients

4 u² + 8 u = 0

ax² + b x + c = 0

a = 4   b = 8   c = 0

Sum of zeroes α + β = -b/a

0 + (-2) = -8/4

- 2 = -2

Product of zeroes α β = c/a

(0)(-2) = 0/4

0 = 0

(v) t² - 15 

Solution :

Find let us find the zeroes of the quadratic polynomial.

t2 - 15 = 0

t2  = 15

  t = √15

  t = ± √15

t = √15   t = - √15

So the values of α = √15 and β = -√15 

Now we are going to verify the relationship between these zeroes and coefficients

t² - 15 = 0

ax² + b x + c = 0

a = 1   b = 0   c = -15

Sum of zeroes α + β = -b/a

√15 + (-√15) = -0/1

0 = 0

Product of zeroes α β = c/a

(√15)( -√15) = -15/1

-15 = -15

(vi) 3 x² – x - 4 

Solution :

Find let us find the zeroes of the quadratic polynomial.

3 x² – x - 4 = 0

(3 x - 4) (x + 1) = 0

(3 x - 4) = 0           (x + 1) = 0

3 x = 4                   x = -1

   x = 4/3          

x = 4/3   x = -1

So the values of α = 4/3 and β = -1

Now we are going to verify the relationship between these zeroes and coefficients

3 x² – x - 4 = 0

ax² + b x + c = 0

a = 3   b = -1   c = -4

Sum of zeroes α + β = -b/a

(4/3) + (-1)  =  -(-1)/3

(4-3)/3  =  1/3

1/3  =  1/3

Product of zeroes α β = c/a

                            (4/3)( -1)  =  -4/3

                               -4/3  =  -4/3

After having gone through the stuff given above, we hope that the students would have understood, how to verify the relation between zeroes and coefficients of the quadratic polynomial.

Apart from the stuff given in this section if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. How to Solve Age Problems with Ratio

    Mar 28, 24 02:01 AM

    How to Solve Age Problems with Ratio

    Read More

  2. AP Calculus BC Integration of Rational Functions by Partical Fractions

    Mar 26, 24 11:25 PM

    AP Calculus BC Integration of Rational Functions by Partical Fractions (Part - 1)

    Read More

  3. SAT Math Practice

    Mar 26, 24 08:53 PM

    satmathquestions1.png
    SAT Math Practice - Different Topics - Concept - Formulas - Example problems with step by step explanation

    Read More