Q. ( z^2 = 4 x + x^2 + 4 x^3 + 4 x^5 ).

Problem

Solve the equation:

\(z^2 = 4x + x^2 + 4x^3 + 4x^5\)

Step 1: Understand what the equation asks

The equation has two variables, \(x\) and \(z\):

\(z^2 = 4x + x^2 + 4x^3 + 4x^5\)

The variable \(z\) is squared. To solve for \(z\), we need to undo the square.

The opposite operation of squaring is taking the square root.

Step 2: Rewrite the right side in standard order

The right side is:

\(4x + x^2 + 4x^3 + 4x^5\)

It is usually easier to read a polynomial when the powers of \(x\) go from highest to lowest.

The highest power is \(x^5\), then \(x^3\), then \(x^2\), then \(x\).

So we can rewrite the equation as:

\(z^2 = 4x^5 + 4x^3 + x^2 + 4x\)

This is the same equation. We only changed the order of the terms.

Step 3: Factor the right side if possible

Now look at the expression:

\(4x^5 + 4x^3 + x^2 + 4x\)

Every term has at least one factor of \(x\):

\(4x^5 = x \cdot 4x^4\)

\(4x^3 = x \cdot 4x^2\)

\(x^2 = x \cdot x\)

\(4x = x \cdot 4\)

So we can factor out \(x\):

\(4x^5 + 4x^3 + x^2 + 4x = x(4x^4 + 4x^2 + x + 4)\)

Now the equation becomes:

\(z^2 = x(4x^4 + 4x^2 + x + 4)\)

Step 4: Check whether the remaining polynomial factors easily

The expression inside the parentheses is:

\(4x^4 + 4x^2 + x + 4\)

There is no common factor among all four terms except \(1\).

We can also check grouping:

\((4x^4 + 4x^2) + (x + 4)\)

The first group has a common factor of \(4x^2\):

\(4x^4 + 4x^2 = 4x^2(x^2 + 1)\)

The second group is:

\(x + 4\)

These groups do not share the same binomial factor, so this grouping does not help.

Therefore, the useful factored form is:

\(z^2 = x(4x^4 + 4x^2 + x + 4)\)

Step 5: Take the square root of both sides

Start with:

\(z^2 = x(4x^4 + 4x^2 + x + 4)\)

Now take the square root of both sides:

\(\sqrt{z^2} = \sqrt{x(4x^4 + 4x^2 + x + 4)}\)

The square root of \(z^2\) gives both a positive and a negative value.

This happens because two numbers can have the same square.

For example:

\(6^2 = 36\)

\((-6)^2 = 36\)

So if \(z^2 = 36\), then \(z = 6\) or \(z = -6\).

Because of this, we must include both signs when solving for \(z\).

So the solution is:

\(z = \pm \sqrt{x(4x^4 + 4x^2 + x + 4)}\)

Step 6: Write the solution as two separate branches

The symbol \(\pm\) means there are two possible expressions for \(z\).

The positive branch is:

\(z = \sqrt{x(4x^4 + 4x^2 + x + 4)}\)

The negative branch is:

\(z = -\sqrt{x(4x^4 + 4x^2 + x + 4)}\)

Step 7: Find the real-number restriction

If we want real-number values of \(z\), the expression inside the square root must be greater than or equal to zero.

So we need:

\(x(4x^4 + 4x^2 + x + 4) \ge 0\)

Now analyze the two factors:

\(x\)

and

\(4x^4 + 4x^2 + x + 4\)

The first factor, \(x\), can be positive, negative, or zero.

The second factor is:

\(4x^4 + 4x^2 + x + 4\)

This expression is always positive for real values of \(x\).

Here is why. The terms \(4x^4\), \(4x^2\), and \(4\) are always nonnegative or positive:

\(4x^4 \ge 0\)

\(4x^2 \ge 0\)

\(4 > 0\)

The only term that can be negative is \(x\). But the positive terms are strong enough to keep the whole expression positive.

To make that clearer, group the expression like this:

\(4x^4 + 4x^2 + x + 4 = 4x^4 + 3x^2 + (x^2 + x + 4)\)

Now look at each part:

\(4x^4 \ge 0\)

\(3x^2 \ge 0\)

For \(x^2 + x + 4\), complete the square:

\(x^2 + x + 4 = \left(x + \frac{1}{2}\right)^2 + \frac{15}{4}\)

Since \(\left(x + \frac{1}{2}\right)^2 \ge 0\) and \(\frac{15}{4} > 0\), we know:

\(\left(x + \frac{1}{2}\right)^2 + \frac{15}{4} > 0\)

Therefore:

\(4x^4 + 4x^2 + x + 4 > 0\)

So the sign of the whole product depends only on \(x\):

\(x(4x^4 + 4x^2 + x + 4) \ge 0\)

Since \(4x^4 + 4x^2 + x + 4\) is always positive, we need:

\(x \ge 0\)

Step 8: Final answer

The equation solved for \(z\) is:

\(z = \pm \sqrt{4x + x^2 + 4x^3 + 4x^5}\)

Equivalently, after factoring the right side:

\(z = \pm \sqrt{x(4x^4 + 4x^2 + x + 4)}\)

For real-number solutions, the restriction is:

\(x \ge 0\)

So the full real-number answer is:

\(z = \sqrt{x(4x^4 + 4x^2 + x + 4)}\) and \(z = -\sqrt{x(4x^4 + 4x^2 + x + 4)}\), where \(x \ge 0\).