# If $\alpha + \beta = 5$ and ${\alpha ^3} + {\beta ^3} = 35$ , find the quadratic equation whose roots are $\alpha $ and $\beta .$

Answer

Verified

380.1k+ views

Hint- To find the quadratic equations first, we have to find the product of roots. We will get it with the help of given values. Then we will put the value of sum of roots and product of roots in quadratic formula.

“Complete step-by-step answer:”

Given that $\alpha + \beta = 5$ and ${\alpha ^3} + {\beta ^3} = 35$

As we know that

$

{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b) \\

{a^3} + {b^3} = {(a + b)^3} - 3ab(a + b) \\

$

We will write this expression in terms of $\alpha $ and $\beta .$

$ \Rightarrow {\alpha ^3} + {\beta ^3} = {(\alpha + \beta )^3} - 3\alpha \beta (\alpha + \beta )$

By putting the value of $\alpha + \beta = 5$ and ${\alpha ^3} + {\beta ^3} = 35$ in above equation, we get

$

\Rightarrow 35 = {(5)^3} - 3\alpha \beta (5) \\

\Rightarrow 35 = 125 - 15\alpha \beta \\

\Rightarrow 15\alpha \beta = 90 \\

\Rightarrow \alpha \beta = 6 \\

$

We know that if $\alpha $ and $\beta $ are the roots quadratic equation, then the quadratic equation is

$ \Rightarrow {x^2} - (\alpha + \beta )x + \alpha \beta = 0$

On substituting the value of $\alpha + \beta = 5$ and $\alpha \beta = 6$ , we get

\[ \Rightarrow {x^2} - 5x + 6 = 0\]

Hence, the quadratic equation will be \[{x^2} - 5x + 6 = 0\]

Note- To solve questions related to quadratic equations, remember the basic properties of quadratic equations such as sum of roots and product of roots of quadratic equation can be used to form the quadratic equations. Root of quadratic equation all satisfies the quadratic equation and some problems this helps to find the coefficients of quadratic equation.

“Complete step-by-step answer:”

Given that $\alpha + \beta = 5$ and ${\alpha ^3} + {\beta ^3} = 35$

As we know that

$

{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b) \\

{a^3} + {b^3} = {(a + b)^3} - 3ab(a + b) \\

$

We will write this expression in terms of $\alpha $ and $\beta .$

$ \Rightarrow {\alpha ^3} + {\beta ^3} = {(\alpha + \beta )^3} - 3\alpha \beta (\alpha + \beta )$

By putting the value of $\alpha + \beta = 5$ and ${\alpha ^3} + {\beta ^3} = 35$ in above equation, we get

$

\Rightarrow 35 = {(5)^3} - 3\alpha \beta (5) \\

\Rightarrow 35 = 125 - 15\alpha \beta \\

\Rightarrow 15\alpha \beta = 90 \\

\Rightarrow \alpha \beta = 6 \\

$

We know that if $\alpha $ and $\beta $ are the roots quadratic equation, then the quadratic equation is

$ \Rightarrow {x^2} - (\alpha + \beta )x + \alpha \beta = 0$

On substituting the value of $\alpha + \beta = 5$ and $\alpha \beta = 6$ , we get

\[ \Rightarrow {x^2} - 5x + 6 = 0\]

Hence, the quadratic equation will be \[{x^2} - 5x + 6 = 0\]

Note- To solve questions related to quadratic equations, remember the basic properties of quadratic equations such as sum of roots and product of roots of quadratic equation can be used to form the quadratic equations. Root of quadratic equation all satisfies the quadratic equation and some problems this helps to find the coefficients of quadratic equation.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Scroll valve is present in a Respiratory system of class 11 biology CBSE

What is BLO What is the full form of BLO class 8 social science CBSE

is known as the Land of the Rising Sun A Japan B Norway class 8 social science CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE