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Question: Find the roots of \[{x^2} - 4ax + 4{a^2} - {b^2} = 0\]...

Find the roots of
x2−4ax+4a2−b2=0{x^2} - 4ax + 4{a^2} - {b^2} = 0

Explanation

Solution

To solve this we have to use the formula that is used to find the value of roots of a quadratic equation. the formula is −b±b2−4ac2a\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} put the values of a,b,ca,b,c in order to get the values of the roots of that equation. while taking out from the root check sign of that term also.

Complete answer: Given,
A quadratic equation
x2−4ax+4a2−b2=0{x^2} - 4ax + 4{a^2} - {b^2} = 0
To find,
The roots of the quadratic equation x2−4ax+4a2−b2=0{x^2} - 4ax + 4{a^2} - {b^2} = 0
To find the roots of this quadratic equation we have to use the formula that is used to find the roots of the general quadratic equation.−b±b2−4ac2a\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}.
Here, bb is the coefficient of xx
aa is the coefficient of x2{x^2}
cc is the constant term of the quadratic equation.
So in the given equation values of a,b,ca,b,c are as follows.
⇒a=1\Rightarrow a = 1,
⇒b=−4a\Rightarrow b = - 4a and
⇒c=4a2−b2\Rightarrow c = 4{a^2} - {b^2}
On putting all these values in the formula of finding the roots of the quadratic equation.
roots=−b±b2−4ac2aroots = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}
roots=−(−4)±(−4a)2−4(1)(4a2−b2)2(1)roots = \dfrac{{ - \left( { - 4} \right) \pm \sqrt {{{\left( { - 4a} \right)}^2} - 4\left( 1 \right)\left( {4{a^2} - {b^2}} \right)} }}{{2\left( 1 \right)}}
On further solving
roots=4±16a2−16a2+4b22roots = \dfrac{{4 \pm \sqrt {16{a^2} - 16{a^2} + 4{b^2}} }}{2}
On simplifying
roots=4±4b22roots = \dfrac{{4 \pm \sqrt {4{b^2}} }}{2}
Now taking 4b24{b^2} outside of bracket
roots=4±2b2roots = \dfrac{{4 \pm 2b}}{2}
On further solving
roots=2±broots = 2 \pm b
First taking a positive sign we get the first root of the equation.
roots=2+broots = 2 + b
Now taking the negative sign we get the second root of the equation.
roots=2−broots = 2 - b
Final answer:
The roots of the quadratic equation:
⇒root=2+b\Rightarrow root = 2 + b and
⇒root=2−b\Rightarrow root = 2 - b
Probability of getting both boys if the elder of them is boy P(Y/X)=12P(Y/X) = \dfrac{1}{2}.

Note:
To solve this type of question we have to use the formula that is used to find the roots of the quadratic equation. then assign the value of a,b,ca,b,c according to the equation. then put all those values in the formula of roots. You may commit a mistake in assigning the value of a,b,ca,b,c and solving the root part of the formula.