Find the step where the circle-equation work goes wrong

Key concept: When a circle equation is being rewritten, the safest move is to check each transformation line by line. One sign error can change the whole result, even if later steps look neat.

Step by step

Start with the given equation:

$x^2 + 4x = -y^2 + 20$

Now compare Kevin’s steps.

Step 1

He rewrites the equation exactly as given:

$x^2 + 4x = -y^2 + 20$

That part is fine.

Step 2

He changes it to

$x^2 + 4x = y^2 + 20$

This is the mistake. The term $-y^2$ should not become $y^2$ unless you multiply the entire equation by $-1$ or move the term properly with opposite sign. He changed only one side’s sign incorrectly.

Quick check

If you wanted to move $-y^2$ to the left side, you would add $y^2$ to both sides:

$x^2 + 4x + y^2 = 20$

That is not what Kevin wrote.

Therefore

The first error happens in Step 2.

Correct choice: 2) Step 2

Pitfall alert

A common trap is to trust the later perfect-looking square completion and ignore the earlier sign change. In algebra, one wrong sign can still lead to a clean-looking but false final form. Always check whether the same operation was applied to both sides.

Try different conditions

If Kevin had instead written

$x^2+4x+y^2=20,$

then he could complete the square on $x$:

$(x+2)^2-4+y^2=20$

which becomes

$(x+2)^2+y^2=24.$

That would describe a circle with center $(-2,0)$ and radius $\sqrt{24}=2\sqrt{6}$.

Further reading

circle equation, completing the square, sign error