   Question Answers

# For a quadratic equation, if one of the root is $\sqrt 2 + \sqrt 3$, then the equation is ${x^2} - 2\sqrt 2 x - 1 = 0$.A. TrueB. False  Hint:Use Shreedharacharya formula to calculate the roots of the quadratic equation. Then check whether one of the roots is$\sqrt 2 + \sqrt 3$.

Formula used:Shreedharacharya formula for finding roots of quadratic equation. $x = \dfrac{{ - b \pm \sqrt D }}{{2a}} = \dfrac{{ - ( - 2\sqrt 2 ) \pm \sqrt {{{( - 2\sqrt 2 )}^2} - 4(1)( - 1)} }}{{2(1)}}$

We are given that a quadratic ${x^2} - 2\sqrt 2 x - 1 = 0$ has one of the roots $\sqrt 2 + \sqrt 3$. We need to check whether the statement is true or not.
To find the roots of the equation we need to use the Shreedharacharya formula. We will start from understanding the Shreedharacharya formula and it’s derivation.
Let the general quadratic equation be $a{x^2} + bx + c = 0$. We have,
$a{x^2} + bx + c = 0$
Multiply both sides by 4a
$\Rightarrow 4{a^2}{x^2} + 4abx + 4ac = 0$
Subtracting both side by 4ac
$\Rightarrow 4{a^2}{x^2} + 4abx = - 4ac$
Adding both sides ${b^2}$ we get,
$\Rightarrow 4{a^2}{x^2} + 4abx + {b^2} = - 4ac + {b^2}$
Completing the square on LHS we get,
$\Rightarrow {(2ax + b)^2} = - 4ac + {b^2} = D$
Taking square roots we get,
$\Rightarrow 2ax + b = \pm \sqrt D .$
Subtracting both sides b and dividing by 2a we get,
$\Rightarrow x = \dfrac{{ - b \pm \sqrt D }}{{2a}}.…………. (1)$
Thus roots of quadratic equation are given by $\Rightarrow x = \dfrac{{ - b \pm \sqrt D }}{{2a}}.$
Now, using the Shreedharacharya formula in ${x^2} - 2\sqrt 2 x - 1 = 0$.
$a = 1$
$b = - 2\sqrt 2$
$c = - 1$
Putting the values of a, b and c in equation (1) we get,
$\Rightarrow x = \dfrac{{ - b \pm \sqrt D }}{{2a}} = \dfrac{{ - ( - 2\sqrt 2 ) \pm \sqrt {{{( - 2\sqrt 2 )}^2} - 4(1)( - 1)} }}{{2(1)}}$.
Simplifying the values we get,
$\Rightarrow x = \dfrac{{2\sqrt 2 \pm \sqrt {8 + 4} }}{2} = \dfrac{{2\sqrt 2 \pm \sqrt {12} }}{2}$
Calculating it further we have,
$\Rightarrow x = \dfrac{{2\sqrt 2 \pm 2\sqrt 3 }}{2}$
Taking 2 common in numerator and cancelling out on the RHS we have,
$\Rightarrow x = \sqrt 2 \pm \sqrt 3$
This shows that the quadratic equation has roots $\sqrt 2 + \sqrt 3$ and $\sqrt 2 - \sqrt 3$.
Since, $\sqrt 2 + \sqrt 3$ is a root of the quadratic equation ${x^2} - 2\sqrt 2 x - 1 = 0$. Therefore, the statement given is true.

So, the correct answer is “Option A”.

Note:While calculating for the roots in a quadratic equation $ax^2+bx+c=0$ we first need to find the D which is also called the discriminant i.e $D=b^2-4ac$. If this discriminant is more than zero then the roots of the equation are real.

CBSE Class 10 Maths Chapter 4 - Quadratic Equations Formula  The Difference Between an Animal that is A Regulator and One that is A Conformer    Solve the Pair of Linear Equation  Where There is a Will There is a Way Essay  Cube Root of 1728  Cube Root of 729  Square Root of 289  How to Find Square Root of a Number  Cube Root of 2  Important Questions for CBSE Class 10 Maths Chapter 4 - Quadratic Equations  Important Questions for CBSE Class 11 Maths Chapter 5 - Complex Numbers and Quadratic Equations  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 10 - A Strange Wrestling Match  Important Questions for CBSE Class 8 Maths Chapter 2 - Linear Equations in One Variable  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 8 - A Pact with the Sun  Important Questions for CBSE Class 6 English A Pact with The Sun Chapter 1 - A Tale of Two Birds    Important Questions for CBSE Class 7 English Honeycomb Chapter 10 - The Story of Cricket  Important Questions for CBSE Class 8 Science Chapter 10 - Reaching The Age of Adolescence  Important Questions for CBSE Class 10 Maths Chapter 3 - Pair of Linear Equations in Two Variables  CBSE Class 10 Hindi A Question Paper 2020  Hindi A Class 10 CBSE Question Paper 2009  Hindi A Class 10 CBSE Question Paper 2015  Hindi A Class 10 CBSE Question Paper 2016  Hindi A Class 10 CBSE Question Paper 2012  Hindi A Class 10 CBSE Question Paper 2010  Hindi A Class 10 CBSE Question Paper 2008  Hindi A Class 10 CBSE Question Paper 2014  Hindi A Class 10 CBSE Question Paper 2007  Hindi A Class 10 CBSE Question Paper 2013  RS Aggarwal Class 10 Solutions - Quadratic Equations  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations  RD Sharma Solutions for Class 10 Maths Chapter 8 - Quadratic Equations  NCERT Solutions for Class 11 Maths Chapter 5  NCERT Exemplar for Class 10 Maths Chapter 4 - Quadratic Equations (Book Solutions)  CBSE Class 8 Maths Chapter 2 Linear Equation in One Variable Exercise 2.5  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations in Hindi  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.4) Exercise 4.4  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.2) Exercise 4.2  NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.3) Exercise 4.3  