Find the zeros of each of the following quadratic polynomial and verify the relationship between the zeros and their coefficients:
(iv) 6x² − 3 − 7x
(v) p(x) = x² + 2√2x - 6
(vi) q(x) = √3x² + 10x + 7v3
Answers
SOLUTION :
(iv) Let f(x) = 6x² −3 − 7x
6x² − 7x −3
By splitting the middle term
= 6x² −7x −3
= 6x² −9x +2x −3
= 3x (2x - 3) +1(2x -3)
= (3x + 1)(2x – 3)
On putting f(x) = 0
(3x + 1)(2x – 3) = 0
(3x + 1)= 0
3x = -1
x = -⅓
(2x – 3) = 0
2x = 3
x = 3/2
Hence, the Zeroes of the polynomials are α = - ⅓ and β = 3/2.
Verification :
Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
−1/3 + 3/2= −(−7)/6
(-1×2 + 3×3)/6 = 7/6
(-2 + 9)/6 = 7/6
7/6 = 7/6
Therefore, Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x².
Product of zeroes (αβ)= constant term /Coefficient of x²
−1/3 × 3/ 2 = −3/6
-1/2 = −1/2
Therefore , Product of zeroes (αβ)= constant term /Coefficient of x²
Hence, the relationship is verified.
(v) p(x) = x² +2√2x – 6
By splitting the middle term
= x² +3√2x - √2x -6
= x(x+3√2) -√2(x+3√2)
= (x+3√2) (x–√2)
On putting p(x) = 0
(x+3√2) (x–√2) = 0
(x+3√2) = 0
x = -3√2
(x–√2) = 0
x = √2
Hence, the Zeroes of the polynomials are α = √2 and β = -3√2.
Verification :
Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x²
√2 - 3√2 = −2√2/1
–2√2 = −2√2
Therefore, Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x².
Product of zeroes (αβ)= constant term /Coefficient of x²
√2× -3√2 = −6/1
-2×3 = -6
-6 = - 6
Therefore ,Product of zeroes (αβ)= constant term /Coefficient of x²
Hence, the relationship is verified.
(vi) q(x) = 3√x² + 10x + 7√3
By splitting the middle term
= 3√x² + 7x+ 3x+7√3
= 3√x(x+√3)+7(x +√3)
= (x +√3)(√3x+ 7)
On putting q(x) = 0
(x +√3)(√3x+ 7) = 0
(x +√3) = 0
x = -√3
(√3x+ 7) = 0
√3x = -7
x = -7/√3
Hence, the Zeroes of the polynomials are α = -√3 and β = -7/√3.
Verification :
Sum of the zeroes(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
Therefore, Sum of the zeroes(α+ β )= −coefficient of x/ coefficient of x².
Product of zeroes (αβ)= constant term /Coefficient of x²
−√3 × –7√3 = 7√3/√3
7 = 7
Therefore, Product of zeroes (αβ)= constant term /Coefficient of x²
Hence, the relationship is verified.
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