Question

find relationship between zeros and coefficient x square + 4 x + 4 is equals to zero​

Answer 1

EXPLANATION.

Relationship between zeroes and coefficient,

⇒ F(x) = x² + 4x + 4 = 0.

As we know that,

Sum of zeroes of a quadratic equation,

⇒ α + β = -b/a.

⇒ α + β = -(4).

Products of zeroes of a quadratic equation,

⇒ αβ = c/a.

⇒ αβ = 4.

Equation = x² + 4x + 4.

Factorizes the equation into middle term splits, we get.

⇒ x² + 2x + 2x + 4 = 0.

⇒ x(x + 2) + 2(x + 2)  = 0.

⇒ (x + 2)(x + 2) = 0.

⇒ (x + 2)² = 0.

⇒ x = -2,-2.

Sum = -2 + (-2) = -4.

Products = (-2)(-2) = 4.

HENCE PROVED.

                                                                                       

MORE INFORMATION.

Conditions for common roots.

Let quadratic equation are a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0.

(1) = If only one root is common.

x = b₁c₂ - b₂c₁/a₁b₂ - a₂b₁.

y = c₁a₂ - c₂a₁/a₁b₂ - a₂b₁.

(2) = if both roots are common,

a₁/a₂ = b₁/b₂ = c₁/c₂.

Answer 2

$\large\underline\purple{\bold{Solution :- }}$

$\tt \ \: :  ⟼ f(x) = {x}^{2} + 4x + 4$

$\tt \ \: :  ⟼ f(x) = {x}^{2} + 2x + 2x + 4$

$\tt \ \: :  ⟼ f(x) = x(x + 2) + 2(x + 2)$

$\tt \ \: :  ⟼ f(x) = (x + 2)(x + 2)$

$\tt\implies \: \: zeroes \: of \: \bf \: f(x) \: \tt \:  = - 2 \: and \: - 2$

$\tt \ \: :  ⟼ Let \: \alpha = - 2 \: and \: \beta = - 2$

$\tt\implies \:Sum \: of \: zeroes \: = \alpha + \beta = - 2 - 2 = - 4$

$\tt\implies \: \alpha + \beta = - \dfrac{4}{1}$

$\tt\implies \: \alpha + \beta = - \dfrac{coefficient \: of \: x}{coefficient \: of \: {x}^{2} }$

$\tt \ \: :  ⟼ Now, \: Product \: of \: zeroes \: = \alpha \beta = ( - 2) \times ( - 2) = 4$

$\tt\implies \: \alpha \beta = 4$

$\tt\implies \: \alpha \beta = \dfrac{4}{1}$

$\tt\implies \: \alpha \beta = \dfrac{constant \: term}{coefficient \: of \: {x}^{2} }$