Chapter 1.4, Problem 55E

Solution

a.

To solve :The equation $x^{2}y+y^2=5$ using the quadratic formula.

The solution of $y=\frac{-x^2+\sqrt{x^4+20}}{2},y=\frac{-x^2-\sqrt{x^4+20}}{2}$ .

Given information:

The given equation is $x^{2}y+y^2=5$ .

Formula used:

Quadratic formula: $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ .

Calculations:

Rewrite the given equation.

  $\begin{array}{l}{x}^{2}y+y^2=5\\ y^2+x^{2}y-5=0\end{array}$

Using the quadratic formula solve for $y$

  $\begin{array}{l}{y}^2+x^{2}y-5=0\\ y=\frac{-x^2\pm \sqrt{x^4+20}}{2}\\ y=\frac{-x^2+\sqrt{x^4+20}}{2},y=\frac{-x^2-\sqrt{x^4+20}}{2}\end{array}$

So the solution of $y=\frac{-x^2+\sqrt{x^4+20}}{2},y=\frac{-x^2-\sqrt{x^4+20}}{2}$ .

b.

To graph: the pair of implicit function $y=\frac{-x^2+\sqrt{x^4+20}}{2},y=\frac{-x^2-\sqrt{x^4+20}}{2}$

Given information:

The given implicit function is . $y=\frac{-x^2+\sqrt{x^4+20}}{2},y=\frac{-x^2-\sqrt{x^4+20}}{2}$

Plot the graph:

The graph of the implicit function $y=\frac{-x^2+\sqrt{x^4+20}}{2}$ is given below:

  

The graph of the implicit function $y=\frac{-x^2-\sqrt{x^4+20}}{2}$ is given below: