Find the zeroes of the quardratic equation 4 x^2-4×1 andverify the relationship between the zeroes and theCofficients. About the author Madeline
Given :- [tex]\sf 4x^{2} -4x + 1 = 0[/tex] To find :- Relationship between the zeroes and the Coefficients. Solution :- We are knowing that α + β = -b/a [tex]\sf \alpha +\beta = \dfrac{-(-4)}{1}[/tex] [tex]\sf \alpha +\beta = \dfrac{4}{1}[/tex] [tex]\alpha +\beta =4[/tex] Product of zeroes [tex]\sf\alpha \beta =\dfrac{c}{a}[/tex] [tex]\sf \alpha \beta =\dfrac{1}{4}[/tex] On factorizing 4x² – 4x + 1 = 0 [tex]\sf 4x^{2} -(2x + 2x) – 1 = 0[/tex] [tex]\sf 4x^{2} – 2x -2x +1 = 0[/tex] [tex]\sf 2x(2x – 1) – 1(2x – 1) = 0.[/tex] Taking 2x – 1 as common [tex]\sf (2x – 1)^2 = 0.[/tex] [tex]\sf x = 1/2[/tex] Sum x = 1/2 + 1/2 x = 1 + 1/2 x = 2/2 x = 1/1 x = 1 Product of zeroes x = 1/2 x 1/2. x = 1 x 1/2 x 2 x = 1/4 Reply
EXPLANATION. Quadratic equation. ⇒ 4x² – 4x + 1 = 0. As we know that, Sum of the zeroes of the quadratic equation. ⇒ α + β = -b/a. ⇒ α + β = -(-4)/4 = 1. Products of the zeroes of the quadratic equation. ⇒ αβ = c/a. ⇒ αβ = 1/4. As we know that, Factorizes the equation into middle term splits, we get. ⇒ 4x² – 4x + 1 = 0. ⇒ 4x² – 2x – 2x + 1 = 0. ⇒ 2x(2x – 1) – 1(2x – 1) = 0. ⇒ (2x – 1)(2x – 1) = 0. ⇒ (2x – 1)² = 0. ⇒ 2x – 1 = 0. ⇒ x = 1/2. Sum of the value of x : ⇒ x = 1/2 + 1/2. ⇒ x = 2/2 = 1. Products of the value of x : ⇒ x = 1/2 x 1/2. ⇒ x = 1/4. Hence verified. MORE INFORMATION. Nature of the factors of the quadratic expression. (1) = Real and different, if b² – 4ac > 0. (2) = Rational and different, if b² – 4ac is a perfect square. (3) = Real and equal, if b² – 4ac = 0. (4) = If D < 0 Roots are imaginary and unequal or complex conjugate. Reply
Given :-
[tex]\sf 4x^{2} -4x + 1 = 0[/tex]
To find :-
Relationship between the zeroes and the Coefficients.
Solution :-
We are knowing that
α + β = -b/a
[tex]\sf \alpha +\beta = \dfrac{-(-4)}{1}[/tex]
[tex]\sf \alpha +\beta = \dfrac{4}{1}[/tex]
[tex]\alpha +\beta =4[/tex]
Product of zeroes
[tex]\sf\alpha \beta =\dfrac{c}{a}[/tex]
[tex]\sf \alpha \beta =\dfrac{1}{4}[/tex]
On factorizing
4x² – 4x + 1 = 0
[tex]\sf 4x^{2} -(2x + 2x) – 1 = 0[/tex]
[tex]\sf 4x^{2} – 2x -2x +1 = 0[/tex]
[tex]\sf 2x(2x – 1) – 1(2x – 1) = 0.[/tex]
Taking 2x – 1 as common
[tex]\sf (2x – 1)^2 = 0.[/tex]
[tex]\sf x = 1/2[/tex]
Sum
x = 1/2 + 1/2
x = 1 + 1/2
x = 2/2
x = 1/1
x = 1
Product of zeroes
x = 1/2 x 1/2.
x = 1 x 1/2 x 2
x = 1/4
EXPLANATION.
Quadratic equation.
⇒ 4x² – 4x + 1 = 0.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ α + β = -(-4)/4 = 1.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ αβ = 1/4.
As we know that,
Factorizes the equation into middle term splits, we get.
⇒ 4x² – 4x + 1 = 0.
⇒ 4x² – 2x – 2x + 1 = 0.
⇒ 2x(2x – 1) – 1(2x – 1) = 0.
⇒ (2x – 1)(2x – 1) = 0.
⇒ (2x – 1)² = 0.
⇒ 2x – 1 = 0.
⇒ x = 1/2.
Sum of the value of x :
⇒ x = 1/2 + 1/2.
⇒ x = 2/2 = 1.
Products of the value of x :
⇒ x = 1/2 x 1/2.
⇒ x = 1/4.
Hence verified.
MORE INFORMATION.
Nature of the factors of the quadratic expression.
(1) = Real and different, if b² – 4ac > 0.
(2) = Rational and different, if b² – 4ac is a perfect square.
(3) = Real and equal, if b² – 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal or complex conjugate.