Q. (y^2 – x^3) (4x) (square)

Problem

Interpret the problem as:

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

This matches the phrase \(4x\) “square,” which means \(4x^2\).

Step 1: Start with the equation

The equation is:

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

We need to solve for \(y\). Since \(y\) is squared, we need to undo the square.

The opposite operation of squaring is taking the square root.

Step 2: Factor the right side

Before taking the square root, simplify the right side if possible.

The right side is:

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

Both terms contain \(x^2\):

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

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

So we can factor out \(x^2\):

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

Now the equation becomes:

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

Step 3: Take the square root of both sides

Start with:

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

Take the square root of both sides:

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

The square root of \(y^2\) gives the absolute value of \(y\):

\(\sqrt{y^2} = |y|\)

So we get:

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

Step 4: Simplify the square root

The expression under the square root is:

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

Since \(x^2\) is a perfect square, we can take it out of the square root:

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

So the equation becomes:

\(|y| = |x|\sqrt{x + 4}\)

Step 5: Remove the absolute value from \(y\)

If \(|y| = |x|\sqrt{x + 4}\), then \(y\) can be positive or negative.

That gives:

\(y = \pm |x|\sqrt{x + 4}\)

Because the \(\pm\) already allows both positive and negative values, this can also be written as:

\(y = \pm x\sqrt{x + 4}\)

Both forms describe the same pair of solution branches.

Step 6: Find the real-number restriction

For \(y\) to be real, the expression inside the square root must be nonnegative.

The square root is:

\(\sqrt{x + 4}\)

So we need:

\(x + 4 \ge 0\)

Subtract \(4\) from both sides:

\(x \ge -4\)

Therefore, real-number solutions exist when:

\(x \ge -4\)

Final answer

The equation solved for \(y\) is:

\(y = \pm x\sqrt{x + 4}\)

So the two solution branches are:

\(y = x\sqrt{x + 4}\)

and

\(y = -x\sqrt{x + 4}\)

For real-number solutions, the restriction is:

\(x \ge -4\)