[[1]] is the square root of 10+91 .

Question

[[1]] is the square root of  10+91 .

Accepted Answer

We are asked to find the square root of

10 + \sqrt{91}

which means the value of

\sqrt{10 + \sqrt{91}}

. Let us assume x and y such that

\sqrt{x} + \sqrt{y}

=

\sqrt{10 + \sqrt{91}}

......(1). Let us square both side of the equations to have;

{(\sqrt{x} + \sqrt{y})}^{2}

=

{(\sqrt{10 + \sqrt{91}})}^{2}

Let us use the algebraic identity of whole square of two numbers (a+b)2=a2+b2+2ab for a=x−−√,b=y√ in the left hand side a of the equation to expand to have;

\Rightarrow x + y + 2 \sqrt{xy} = 10 + \sqrt{91} . . . . . (2)

Let us compare the rational part of the above equation and have;
x+y=10......(3)
Let us compare the irrational part of the above equation(2) and have;

2 \sqrt{xy} = \sqrt{91}

\Rightarrow \sqrt{xy} = \frac{\sqrt{91}}{2}

We square both sides of above equation to have;

xy = \frac{91}{4}

\Rightarrow x = \frac{91}{4 y}

We put the value of x in equation (3) to have;
x+y=10

\Rightarrow \frac{91}{4 y} + y = 10

\Rightarrow \frac{91 + 4 y^{2}}{4 y} = 10

\Rightarrow 4 y^{2} - 40 y + 91 = 0

We solve the above quadratic equation in y by splitting the middle term method.
⇒4y2−26y−14y+91=0
⇒2y(2y−13)−7(2y−13)=0
⇒(2y−13)(2y−7)=0
⇒2y−13=0 or 2y−7=0
So we have two solutions for y as