Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
(iv)
(v)
(vi)
Answers
SOLUTION :
(iv) Given : f(x) = 6x² − 3− 7x
f(x) = 6x² −7x− 3
f(x)= 6x² - 9x + 2x -3
f(x)= 3x ( 2x - 3) + 1 (2x - 3)
f(x) = (3x + 1)(2x– 3)
To find zeroes, put f(x) = 0
(3x + 1)(2x– 3) = 0
(3x + 1) = 0 or (2x– 3) = 0
3x = -1 or 2x = 3
x = -1/3 or x = 3/2
Hence, Zeroes of the polynomials are α = - ⅓ and β = 3/2
VERIFICATION :
Sum of the zeroes = −coefficient of x / coefficient of x²
α + β = −coefficient of x / coefficient of x²
−1/3 + 3/2 = −(−7)/6
(-2+ 9)/6 = 7/6
7/6 = 7/6
Product of the zeroes = constant term/ Coefficient of x²
α β = constant term / Coefficient of x²
−1/3 × 3/2 = −3/6
−3/6 = −3/6
-½ = -½
Hence, the relationship between the Zeroes and its coefficients is verified.
(v) GIVEN : p(x)= x² + 2√2x - 6
= x² + 3√2x + √2x - 6
= x(x + 3√2)–√2(x + 3√2)
= (x + 3√2)(x–√2)
To find zeroes, put p(x) = 0
(x + 3√2)(x–√2) = 0
(x + 3√2) = 0 or (x–√2) = 0
x = -3√2 or x = √2
Hence, Zeroes of the polynomials are α = -3√2 and β = √2
VERIFICATION :
Sum of the zeroes = − coefficient of x / coefficient of x²
α + β = −coefficient of x / coefficient of x²
−3√2 + √2 = −2√2/1
√2 (-3 +1) = −2√2/1
√2 × - 2 = −2√2/1
-2√2 = -2√2
Product of the zeroes = constant term/ Coefficient of x²
α β = constant term / Coefficient of x²
−3√2 × √2 = −6/1
-3× 2 = -6
-6 = -6
Hence, the relationship between the Zeroes and its coefficients is verified.
(vi) Given : q(x) = √3x² + 10x + 7√3
= √3x² + 3x + 7x + 7√3
= √3x(x + √3 ) + 7 ( x + √3)
[√3 ×√3 = 3]
= (x + √3) (√3x +7)
To find zeroes, put q(x) = 0
(x + √3) (√3x +7) = 0
(x + √3) = 0 or (√3x +7) = 0
x = - √3 or x = -7/√3
Hence, Zeroes of the polynomials are α = - √3 and β = -7/√3
VERIFICATION :
Sum of the zeroes = − coefficient of x / coefficient of x²
α + β = −coefficient of x / coefficient of x²
-√3 + -7/√3 = − (10) / √3
[(-√3 × √3 ) -7 ]/√3 = − (10) / √3
(-3 - 7) /√3 = − 10 / √3
-10/√3 = − 10 / √3
Product of the zeroes = constant term/ Coefficient of x²
α β = constant term / Coefficient of x²
-√3 × -7/√3 = 7√3 /√3
7 = 7
Hence, the relationship between the Zeroes and its coefficients is verified.
HOPE THIS ANSWER WILL HELP YOU….
ur answer
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Given quadratic equation = √3x² - 8x + 4√3 = 0
We should factorize the equation first.
=√3x²-6x-2x+4√3 = 0
= √3x(x-2√3)-2(x-2√3) = 0
= (√3x-2) (x-2√3) = 0
= (√3x-2) = 0 , (x-2√3) = 0
= x = 2/√3 , x = 2√3.
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