Question

x 2 - 4ax + 4a 2 - b 2 =0 find the roots of this quadratic equation by method of completing the square.

Answer 1

Answer:

$Roots \:of\\\: given\: quadratic\: equation\: are\: ,\\x = 2a+b \: Or \: x = 2a-b$

Step-by-step explanation:

Given quadratic equation:

x²-4ax+4a²-b²=0

Finding roots of the quadratic equation By completing the square method:

Rearranging the equation, we get

=> x²-4ax+4a² = b²

$x^{2}-2\times x \times 2a +(2a)^{2}=b^{2}$

$\implies (x-2a)^{2}=b^{2}$

$\implies x-2a = ±\sqrt{b^{2}}$

$\implies x-2a=±b$

$\implies x = 2a±b$

$\implies x = 2a+b \: Or \: x = 2a-b$

Therefore,

$x = 2a+b \: Or \: x = 2a-b$

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Answer 2

(2a-b)  and (2a+b) are the factors of the quadratic equation  $\bold{x^{2}-4 a x+4 a^{2}-b^{2}=0}$ by the method of completing the square.

Given:  

$x^{2}-4 a x+4 a^{2}-b^{2}=0$

To find:

Factors of $x^{2}-4 a x+4 a^{2}-b^{2}=0$  by completing the square method

Solution:

$\begin{array}{l}{x^{2}-4 a x+4 a^{2}-b^{2}=0} \\ {(x)^{2}+2(-2 a) x+(2 a)^{2}-b^{2}=0} \\ {(x-2 a)^{2}-b^{2}=0} \\ {(x-2 a+b)(x-2 a-b)=0}\end{array}$

Either,

$\begin{array}{l}{(x-2 a+b)=0} \\ {x=2 a-b} \\ {o r} \\ {(x-2 a-b)=0} \\ {x=2 a+b}\end{array}$

Thus, the factors of $\bold{x^{2}-4 a x+4 a^{2}-b^{2}=0}$ are found to be $\bold{(2a-b) and (2a+b).}$