Question

Two root of quadratic equation are given:Frame the equation. 1-3√5,1+3√5​

Answer 1

Answer:x²-2x-44=0

Step-by-step explanation:Two roots are given

S=1-3√5+1+3√5=2

P=(1-3√5)(1+3√5)

Therefore, P=(1)^2-(3√5)^2

P=1-45=-44

x²-Sx+P=0

x²-2x-44=0

Answer 2

Answer:

The required quadratic equation is

$\boxed{\red{\sf\:x^{2}\:-\:2x\:-\:44\:=\:0}}$

Step-by-step-explanation:

We have given that, two roots of a quadratic equation are

$\sf\:1\:-\:3\:\sqrt{5}\:\:\&\:\:1\:+\:3\:\sqrt{5}$.

Let the roots be

$\sf\:\alpha\:\&\:\beta\\\\\therefore\sf\:\alpha\:=\:1\:-\:3\:\sqrt{5}\:\:\&\:\:\\\\\sf\:\beta\:=\:1\:+\:3\:\sqrt{5}$.

Now,

$\red{\sf\:Sum\:of\:roots\:=\:\alpha\:+\:\beta}\\\\\sf\:Sum\:of\:roots\:=\:1\:-\cancel{\:3\:\sqrt{5}}\:+\:1\:+\cancel{\:3\:\sqrt{5}}\\\\\sf\:Sum\:of\:roots\:=\:1\:+\:1\\\\\boxed{\red{\sf\:Sum\:of\:roots\:=\:2}}\\\\\sf\:Now,\\\\\pink{\sf\:Product\:of\:roots\:=\:\alpha.\:\beta}\\\\\sf\:Product\:of\:roots\:=\:(\:1\:-\:3\:\sqrt{5}\:)\:\times\:(\:1\:+\:3\:\sqrt{5}\:)\\\\\sf\:Product\:of\:roots\:=\:(\:1\:)^{2}\:-\:(\:3\:\cancel{\sqrt{5}}\:)^{\cancel{2}}\:\:\:[\:(\:a\:-\:b\:)\:(\:a\:+\:b\:)\:=\:a^{2}\:-\:b^{2}\:]\\\\\sf\:Product\:of\:roots\:=\:1\:-\:9\:\times\:5\\\\\sf\:Product\:of\:roots\:=\:1\: - \:45\\\\\boxed{\pink{\sf\:Product\:of\:roots\:=\:-\:44}}$

Now, the required quadratic equation is in the form

$\red{\sf\:x^{2}\:-\:(\:\alpha\:+\:\beta\:)\:x\:+\:(\:\alpha.\:\beta\:)\:=\:0}\\\\\implies\sf\:x^{2}\:-\:(\:2\:)\:x\:+\:(\:-\:44\:)\:=\:0\\\\\implies\boxed{\red{\sf\:x^{2}\:-\:2x\:-\:44\:=\:0}}$

Additional Information:

1. Quadratic Equation :

An equation having a degree '2' is called quadratic equation.

The general form of quadratic equation is

$\sf\:ax^{2}\:+\:bx\:+\:c\:=\:0$

Where, a, b, c are real numbers and a ≠ 0.

2. Roots of Quadratic Equation:

The roots means nothing but the value of the variable given in the equation.

3. Methods of solving quadratic equation:

There are mainly three methods to solve or find the roots of the quadratic equation.

A) Factorization method

B) Completing square method

C) Formula method

4. Formula for solving a quadratic equation:

$\boxed{\red{\sf\:x\:=\:\dfrac{\:-\:b\:\pm\:\sqrt{b^{2}\:-\:4ac}}{2a}}}$