Question

Form the quadratic equation from the roots given below.

2 - √5, 2 + √5





please give the answer fast friends please help me fast ​

Answer 1

GIVEN:

• One of roots of quadratic equation are 2 - √5 & 2 + √5.

To Find:

• Quadratic equation.

Solution:

let us consider here,

$\bf\alpha{ = 2 - \sqrt{5} }$

$\bf\beta{ = 2 + \sqrt{5} }$

Now,

$\star \: {\sf\green{Sum \: of \: roots = \alpha + \beta}}$

$\bf \implies \alpha\ + \beta =\ (2\ -\ \sqrt{5})\ + (2\ +\ \sqrt{5})$

$\bf:\implies{\alpha + \beta = 2 + 2}$

$\sf\star \: \underbrace{\alpha + \beta = 4} \: \star$

$\star \: {\sf\green{Product \: of \: roots = \alpha \times \beta}}$

$\bf \implies \alpha\ \times\ \beta =\ (2\ -\ \sqrt{5})\ \times\ (2\ +\ \sqrt{5})$

$\bf:\implies{\alpha \times \beta =4 - 5}$

$\sf\star \: \underbrace{\alpha \times \beta = - 1} \: \star$

We know that,

$\star \: {\sf\green{Quadratic \: Equation = {x }^{2} - (\alpha + \beta)x + (\alpha \times \beta = 0}}$

$\bf:\implies{ {x}^{2} - 4 \times x - 1 = 0 }$

$\sf\star \: \underbrace{ {x}^{2} - 4x - 1 = 0 } \: \star$

Hence,

• Quadratic Equation is x² - 4x - 1 = 0.

Answer 2

they are roots 2 - √5, 2 + √5 then

sum of roots= a+b =( 2 - √5) +( 2 + √5) = 4

product of roots = a×b = (2 - √5)×(2 + √5)

= 2²-√5²= 4 -5= -1

standard form is x² -(a+b)x + ab =0

x² -4x -1 =0