Question

Find the value of $\sqrt{[-3+\sqrt{\\{-3+\sqrt{-3+.....\infty \\}]}}}$

Answer 1

We have to find the value of the given term. As we can see that it is an infinite value so firstly we will put it equal to any variable then as they are square root values we will square both sides and take all the terms without the square root on one side and the rest on another. Finally by using the value we have let we will solve the obtained equation and get the desired answer.

Complete step-by-step solution:
We have to find the value of the below number:
$\sqrt{[-3+\sqrt{\\{-3+\sqrt{-3+.....\infty \\}]}}}$
Let us take the above value equal to a variable $X$ as follows:
$X=\sqrt{[-3+\sqrt{\\{-3+\sqrt{-3+.....\infty \\}]}}}$…..$\left( 1 \right)$
On squaring both sides we get,
${{X}^{2}}=-3+\sqrt{-3+\sqrt{-3+......\infty }}$
Now we will take all term without square root on one side as follows:
${{X}^{2}}+3=\sqrt{-3+\sqrt{-3+\sqrt{-3......\infty }}}$
Substituting value from equation (1) above we get,
${{X}^{2}}+3=X$
$\Rightarrow {{X}^{2}}-X+3=0$…..$\left( 2 \right)$
Now using the Quadratic Formula we will solve the above equation.
Quadratic formula for quadratic equation
$a{{x}^{2}}+bx+c=0$….$\left( 3 \right)$
Is given as:
$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ ….$\left( 4 \right)$
On comparing equation (2) and (3) we get,
$a=1$ $b=-1$ and $c=3$
Substitute the above value in equation (4) and solve as follows:
$X=\dfrac{-\left( -1 \right)\pm \sqrt{{{\left( -1 \right)}^{2}}-4\times 1\times 3}}{2\times 1}$
$X=\dfrac{1\pm \sqrt{1-12}}{2}$
On solving further we get,
$X=\dfrac{1\pm \sqrt{-11}}{2}$
$X=\dfrac{1\pm \sqrt{11}i}{2}$
On substituting value from equation (1) above we get,
$\sqrt{[-3+\sqrt{\\{-3+\sqrt{-3+.....\infty \\}]}}}=\dfrac{1\pm \sqrt{11}i}{2}$
Hence value of $\sqrt{[-3+\sqrt{\\{-3+\sqrt{-3+.....\infty \\}]}}}$ is $\dfrac{1\pm \sqrt{11}i}{2}$.

Note: Quadratic equations are those equations whose highest power of the variable is two. It has only one unknown variable with highest power as two. All the value of the variable that satisfies the equation is known as the solution of the equation. Quadratic equations have at-most two solutions. There are many ways to solve a quadratic equation such as completing the square, Discriminant or using Geometric Interpretation.