
If $x = \dfrac{2}{3}$ and x = -3 are the roots of the equation $a{x^2} + 7x + b = 0,$ find the values of ${a^2} + {b^2}.$
Answer
549.9k+ 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
Master Class 10 Computer Science: Engaging Questions & Answers for Success

Master Class 10 General Knowledge: Engaging Questions & Answers for Success

Master Class 9 General Knowledge: Engaging Questions & Answers for Success

Master Class 9 English: Engaging Questions & Answers for Success

Master Class 9 Maths: Engaging Questions & Answers for Success

Master Class 9 Social Science: Engaging Questions & Answers for Success

Trending doubts
Which one is a true fish A Jellyfish B Starfish C Dogfish class 10 biology CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Why is there a time difference of about 5 hours between class 10 social science CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Write examples of herbivores carnivores and omnivo class 10 biology CBSE

When and how did Canada eventually gain its independence class 10 social science CBSE
