Question

OPEN CHALLENGE :
The sum of the two roots of a quadratic equation is 5 & sum of their cubes is 35. Find the equation.

Answer 1

hey friend here is your answer _____________ plz see this attachment
hope this helps you...
if u want the zeroes then plz see the 2nd pic.

Answer 2

Answer:

The required quadratic equation is

$\boxed{\red{\sf\:x^2\:-\:5x\:+\:6\:=\:0}}$

Step-by-step-explanation:

We have given that,

The sum of the roots of a quadratic equation is 5.

And the sum of the cubes of roots is 35.

We have to find the quadratic equation.

Let the roots of the quadratic equation be $\sf\:\alpha\:\&\:\beta$.

From the first condition,

$\sf\:\alpha\:+\:\beta\:=\:5\:\:\:\:-\:-\:(\:1\:)$

From the second condition,

$\sf\:\alpha^3\:+\:\beta^3\:=\:35\:\:\:-\:-\:(\:2\:)$

Now, we know that,

$\sf\:\alpha^3\:+\:\beta^3\:=\:(\:\alpha\:+\:\beta\:)^3\:-\:3\:\alpha\:\beta\:(\:\alpha\:+\:\beta\:)\:\:\:-\:-\:[\:Identity\:]\\\\\\\implies\sf\:\alpha^3\:+\:\beta^3\:=\:(\:5\:)^3\:-\:3\:\alpha\:\beta\:(\:5\:)\:\:\:-\:-\:[\:From\:(\:1\:)\:]\\\\\\\implies\sf\:\alpha^3\:+\:\beta^3\:=\:125\:-\:15\:\alpha\:\beta\\\\\\\implies\sf\:35\:=\:125\:-\:15\:\alpha\:\beta\:\:-\:-\:[\:From\:(\:2\:)\:]\\\\\\\implies\sf\:15\:\alpha\:\beta\:=\:125\:-\:35\\\\\\\implies\sf\:15\:\alpha\:\beta\:=\:90\\\\\\\implies\sf\:\alpha\:.\:\beta\:=\:\cancel{\dfrac{90}{15}}\\\\\\\implies\boxed{\red{\sf\:\alpha\:.\:\beta\:=\:6}}$

Now,

The required quadratic equation is in the form,

$\pink{\sf\:x^2\:-\:(\:\alpha\:+\:\beta\:)\:x\:+\:\alpha\:.\:\beta\:=\:0}\\\\\\\implies\sf\:x^2\:-\:(\:5\:)\:x\:+\:(\:6\:)\:=\:0\\\\\\\implies\boxed{\red{\sf\:x^2\:-\:5x\:+\:6\:=\:0}}$

Additional Information:

1. Quadratic Equation:

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

The general form of quadratic equation is

ax² + 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 to solve quadratic equation:

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