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Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81?
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Find the zeros of the following quadratic polynomias and verify the re...
Answer :
(a). The given polynomial is 6x²-x-1.
Therefore, 6x²-x-1=0
6x²-3x+2x-1=0
3x(2x-1)+1(2x-1)=0
(2x-1)(3x+1)=0
x= 1/2 , -1/3.
So , the zeroes are 1/2 and -1/3.
Since the given equation is 6x²-x-1.
On compairing this wit ax²+bx+c=0
we get, a=6 , b=-1 , c=-1.
Now , sum of the zeroes = -b/a
1/2 - 1/3 = 1/6
( 3-2)/6 = 1/6
1/6 = 1/6
R.H.S = L.H.S Verified.
and product of zeroes = c/a
1/2×(-1/3) = -1/6
-1/6 = -1/6
L.H.S = R.H.S Verified.
(b). The given polynomial is 48y²-13y-1=0
Therefore , 48y²-16y+3y-1=0
16y(3y-1)+1(3y-1)=0
(3y-1)(16y+1)=0
y= 1/3 , -1/16
So, the zeroes are 1/3 and -1/16.
Since the given equation is 48y²-13y-1
On compairing this with ay²+by+c=0
we get, a=48 , b=-13 , c=-1
Now , Sum of zeroes = -b/a
1/3 - 1/16 = 13/48
(16-3)/48 = 13/48
13/48 = 13/48
L.H.S = R.H.S Verified .
and product of zeroes = c/a
1/3 × (-1/16) = -1/48
-1/48 = -1/48 Verified.
(c). The given equation is 63-2x-x² =0
Therefore , x²+2x-63 = 0
x²+9x-7x-63 = 0
x(x+9) -7(x+9) = 0
(x+9)(x-7) = 0
x= -9 , 7
So, the zeroes are -9 and 7
Since the given equation is x²+2x-63=0
On compairing this with ax²+bx+c=0
we get , a=1 , b=2 and c=-63
Now, Sum of zeroes = -b/a
-9+7 = -2/1
-2 = -2
L.H.S = R.H.S Verified.
and product of zeroes = c/a
-9 × 7 = -63/1
-63 = -63
L.H.S = R.H.S Verified.
(d). The given equation is 49x²-81=0
Therefore, (7x)² - (9)² =0
(7x+9)(7x-9) = 0
x = -9/7 , 9/7
So, the zeroes are -9/7 and 9/7.
Since the given equation is 49x²-81=0
On compairing this with ax²+bx+c=0
we get, a=49 , b=0 and c=-81
Now, Sum of zeroes = -b/a
-9/7+9/7 = -0/49
0 = 0
L.H.S = R.H.S Verified.
and product of zeroes = c/a
-9/7 × 9/7 = -81/49
-81/49 = -81/49
L.H.S = R.H.S Verified.
Hope its help you.
(a). The given polynomial is 6x²-x-1.
Therefore, 6x²-x-1=0
6x²-3x+2x-1=0
3x(2x-1)+1(2x-1)=0
(2x-1)(3x+1)=0
x= 1/2 , -1/3.
So , the zeroes are 1/2 and -1/3.
Since the given equation is 6x²-x-1.
On compairing this wit ax²+bx+c=0
we get, a=6 , b=-1 , c=-1.
Now , sum of the zeroes = -b/a
1/2 - 1/3 = 1/6
( 3-2)/6 = 1/6
1/6 = 1/6
R.H.S = L.H.S Verified.
and product of zeroes = c/a
1/2×(-1/3) = -1/6
-1/6 = -1/6
L.H.S = R.H.S Verified.
(b). The given polynomial is 48y²-13y-1=0
Therefore , 48y²-16y+3y-1=0
16y(3y-1)+1(3y-1)=0
(3y-1)(16y+1)=0
y= 1/3 , -1/16
So, the zeroes are 1/3 and -1/16.
Since the given equation is 48y²-13y-1
On compairing this with ay²+by+c=0
we get, a=48 , b=-13 , c=-1
Now , Sum of zeroes = -b/a
1/3 - 1/16 = 13/48
(16-3)/48 = 13/48
13/48 = 13/48
L.H.S = R.H.S Verified .
and product of zeroes = c/a
1/3 × (-1/16) = -1/48
-1/48 = -1/48 Verified.
(c). The given equation is 63-2x-x² =0
Therefore , x²+2x-63 = 0
x²+9x-7x-63 = 0
x(x+9) -7(x+9) = 0
(x+9)(x-7) = 0
x= -9 , 7
So, the zeroes are -9 and 7
Since the given equation is x²+2x-63=0
On compairing this with ax²+bx+c=0
we get , a=1 , b=2 and c=-63
Now, Sum of zeroes = -b/a
-9+7 = -2/1
-2 = -2
L.H.S = R.H.S Verified.
and product of zeroes = c/a
-9 × 7 = -63/1
-63 = -63
L.H.S = R.H.S Verified.
(d). The given equation is 49x²-81=0
Therefore, (7x)² - (9)² =0
(7x+9)(7x-9) = 0
x = -9/7 , 9/7
So, the zeroes are -9/7 and 9/7.
Since the given equation is 49x²-81=0
On compairing this with ax²+bx+c=0
we get, a=49 , b=0 and c=-81
Now, Sum of zeroes = -b/a
-9/7+9/7 = -0/49
0 = 0
L.H.S = R.H.S Verified.
and product of zeroes = c/a
-9/7 × 9/7 = -81/49
-81/49 = -81/49
L.H.S = R.H.S Verified.
Hope its help you.
Community Answer
Find the zeros of the following quadratic polynomias and verify the re...
Finding Zeros of Quadratic Polynomials
Quadratic Polynomials
A quadratic polynomial is a polynomial of degree 2, which means that the highest exponent of the variable is 2. A quadratic polynomial can be written in the form ax^2 + bx + c, where a, b, and c are constants.
Finding Zeros of Quadratic Polynomials
The zeros of a quadratic polynomial are the values of x that make the polynomial equal to zero. These values can be found by using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/2a
To use this formula, we need to know the values of a, b, and c in the quadratic polynomial.
Relationship between Zeros and Coefficients
If the zeros of a quadratic polynomial are x1 and x2, then the polynomial can be written as:
a(x - x1)(x - x2) = ax^2 - a(x1 + x2)x + ax1x2
Comparing this with the standard form of a quadratic polynomial, we can see that:
a = leading coefficient
-(x1 + x2) = coefficient of x
x1x2 = constant term
Therefore, the sum of the zeros is equal to the negative of the coefficient of x, divided by the leading coefficient, and the product of the zeros is equal to the constant term, divided by the leading coefficient.
Examples
Example (a)
Polynomial: 6x^2 - x - 1
Using the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4(6)(-1)))/2(6)
x1 = 1/2
x2 = -1/3
Sum of zeros = (1/2) + (-1/3) = 1/6
Product of zeros = (1/2)(-1/3) = -1/6
Coefficient of x = -1
Relationship between zeros and coefficients is verified.
Example (b)
Polynomial: 48y^2 - 13y - 1
Using the quadratic formula:
y = (-(-13) ± √((-13)^2 - 4(48)(-1)))/2(48)
y1 = 1/6
y2 = -1/8
Sum of zeros = (1/6) + (-1/8) = 5/24
Product of zeros = (1/6)(-1/8) = -1/48
Coefficient of y = -13
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Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81?
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Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81?.
Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the zeros of the following quadratic polynomias and verify the relationship between the zeros and The r coefficients: (a)6x^2-x-1 (b)48y^2-13y-1 (c)63-2x-x^2 (d)49x^2-81?.
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