Question

Find the zeroes of the quadratic polynomial 9x2 – 6x + 1 and verify the relationship between the zeroes and the coefficients​

Answer 1

Step-by-step explanation:

Given:-

The quadratic polynomial 9x^2 – 6x + 1

To find:-

Find the zeroes of the quadratic polynomial

9x^2 – 6x + 1 and verify the relationship between the zeroes and the coefficients

Solution:-

Given quadratic polynomial is 9x^2-6x+1

=>P(x)= 9x^2 - 6x +1

=>P(x)= 9x^2 -3x -3x +1

=>P(x) = 3x ( 3x -1) -1( 3x-1)

=>P(x) = (3x - 1) (3x-1)

To get the zeores we can write P(x) = 0

=> P(x) = (3x - 1) (3x-1) = 0

=> 3x -1 = 0

=> 3x = 1

=> x = 1/3

The zeores are 1/3 and 1/3

Let α = 1/3 and β = 1/3

P(x)=9x^2 - 6x +1

On Comparing this with the standard quadratic Polynomial ax^2+bx+c

a = 9

b=-6

c=1

Sum of the zeros =

α + β = (1/3)+(1/3)

=> (1+1)/3

=> 2/3

=-(-6/9)

=> -b/a

α + β = b/a

Product of the zeroes = α β

α + β = (1/3)(1/3)

=> 1/9

=> c/a

α β = c/a

Verified the given relations between the zeroes and the coefficients of the given Polynomial.

Used formulae:-

• Sum of the zeros = -b/a

• Product of the zeroes = c/a

Answer 2

Given :-

A quadratic polynomial 9x² - 6x + 1 .

To Find :-

The zeroes of the given polynomial and verify the relationship between the zeroes and the coefficients respectively .

Used Concepts :-

• A general quadratic polynomial is in the form of "ax² + bx + c" .
• The sum of zeroes of a quadratic polynomial is given by "-b/a".
• The product of Zeroes of a quadratic polynomial is given by "c/a".

Solution :-

Let , p(x) = 9x² - 6x + 1

For Zeroes p(x) = 0

9x² - 6x + 1 = 0

9x² - 3x - 3x + 1 = 0

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

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

Either , 3x - 1 = 0 or 3x - 1 = 0

3x - 1 = 0 , 3x - 1 = 0

3x = 1 , 3x = 1

x = 1/3 , x = 1/3

Therefore , both zeroes are equal . So , the Discrimanant of the given polynomial is "0".

Now , Sum of roots = 1/3 + 1/3 = 2/3

Sum of roots by "-b/a" :-

The polynomial = 9x² - 6x + 1

Sum of roots = -b/a = -(-6)/9 = 6/9 = 2/3

Product of roots = 1/3 × 1/3 = 1/9

Product of roots by "c/a" :-

The polynomial = 9x² - 6x + 1

Product of roots = c/a = 1/9 .

Therefore , Verified !

Additional Information :-

• A general Quadratic equation is in the form of "ax² + bx + c = 0".
• Sum of roots of a quadratic equation is given by "-b/a".
• Product of roots of a quadratic equation is given by "c/a".
• If D = 0 is equal to zero , Then The roots are equal .
• If D < 0 , Then the roots are imaginary and distinct .
• If D > 0 , Then two real and distinct roots exist here.
• If D is a perfect square number then the roots are rational .
• The graph of a quadratic equation is parabola.
• The vertex of the parabola is given by [ -b/2a , -D/4a ] .
• If the coefficient of x² is negative then the parabola opens downwards .
• If the coefficient of x² is positive then the parabola opens upwards .
• If the quadratic equation is in variable "x" and The parabola is not cutting the " X - axis" anywhere . So the roots are imaginary.
• If the quadratic equation is in variable "x" and the parabola is touching the "X - axis " at only one point . Then the roots are equal and the discriminant ( D ) is "0".
• D = b² - 4ac .
• Quadratic formulae = -b + √D/2a , -b - √D/2a .
• A general cubic equation is in the form of ax³ + bx² + cx + d = 0 .
• The graph of a cubic equation is a curve .
• If ( a , b and c ) are roots of a cubic Equation . Then ,

1. a + b + c = -b/a.
2. ab + bc + ca = c/a.
3. a × b × c = -d/a.

Thank You !