If $x = \dfrac{2}{3}$ and x = -3 are the roots of the equation $a{x^2} + 7x + b = 0,$ find the values of ${a^2} + {b^2}.$
Last updated date: 22nd Mar 2023
•
Total views: 306.6k
•
Views today: 7.85k
Answer
306.6k+ views
Hint: Factorization of a quadratic equation gives us roots. Here the roots are given. We can follow the reverse process to find the required quadratic equation and then compare it with the given equation to get coefficients a, b.
Complete step-by-step answer:
The given quadratic equation is $a{x^2} + 7x + b = 0$
The roots of this equation are given as $x = \dfrac{2}{3}\;\& \;x = - 3$.
Then we can write as
$\left( {x - \dfrac{2}{3}} \right)\left( {x - ( - 3)} \right) = 0$ It will be the quadratic equation found from the given roots.
$ \Rightarrow \left( {x - \dfrac{2}{3}} \right)\left( {x + 3} \right) = 0$
On simplification of the above equation,
$ \Rightarrow {x^2} + 3x - \dfrac{{2x}}{3} - \dfrac{{2 \times 3}}{3} = 0$
$ \Rightarrow {x^2} + \dfrac{7}{3}x - 2 = 0$
Multiplying the above equation with 3 on both sides, we get
$ \Rightarrow 3{x^2} + 7x - 6 = 0$
Comparing the above equation with the given quadratic equation $a{x^2} + 7x + b = 0,$ we get
a = 3, b = –6.
$ \Rightarrow {a^2} + {b^2} = {(3)^2} + {( - 6)^2} = 9 + 36 = 45$
$\therefore $ The value of ${a^2} + {b^2} = 45$
Note: We can use a different way to solve the given problem using sum of the roots and product of the roots. Standard form of a quadratic equation with roots a, b can be written as
${x^2} - (sum\;of\;roots)x + product\;of\;roots = 0$
${x^2} - (a + b)x + ab = 0$
Complete step-by-step answer:
The given quadratic equation is $a{x^2} + 7x + b = 0$
The roots of this equation are given as $x = \dfrac{2}{3}\;\& \;x = - 3$.
Then we can write as
$\left( {x - \dfrac{2}{3}} \right)\left( {x - ( - 3)} \right) = 0$ It will be the quadratic equation found from the given roots.
$ \Rightarrow \left( {x - \dfrac{2}{3}} \right)\left( {x + 3} \right) = 0$
On simplification of the above equation,
$ \Rightarrow {x^2} + 3x - \dfrac{{2x}}{3} - \dfrac{{2 \times 3}}{3} = 0$
$ \Rightarrow {x^2} + \dfrac{7}{3}x - 2 = 0$
Multiplying the above equation with 3 on both sides, we get
$ \Rightarrow 3{x^2} + 7x - 6 = 0$
Comparing the above equation with the given quadratic equation $a{x^2} + 7x + b = 0,$ we get
a = 3, b = –6.
$ \Rightarrow {a^2} + {b^2} = {(3)^2} + {( - 6)^2} = 9 + 36 = 45$
$\therefore $ The value of ${a^2} + {b^2} = 45$
Note: We can use a different way to solve the given problem using sum of the roots and product of the roots. Standard form of a quadratic equation with roots a, b can be written as
${x^2} - (sum\;of\;roots)x + product\;of\;roots = 0$
${x^2} - (a + b)x + ab = 0$
Recently Updated Pages
If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE
