Negative Strips and Missing Area

Begin with a positive strip

Begin with x squared plus six x plus five equals zero. Each strip has width three, and the full corner is nine. Five units are already present. This familiar positive example gives us a reference before we change any signs.

Add the missing four to both sides. The completed square has side expression x plus three and area four. The tiles show signed algebraic contributions. Their drawn dimensions are schematic because x is still an unknown.

The positive square-root branch gives two. Subtract the strip width three, and x is negative one. Keep that answer in mind as we reverse the linear term next.

The other branch gives negative two minus three, or negative five. Changing a branch preserves the coefficients. We are comparing two solutions of one equation, before moving to a different equation.

Reverse the strips

Change six x to negative six x. The two strips now each contribute negative three x. Their signed width is negative three. Their physical drawing does not acquire a negative size; the sign is carried by the labels and the signed-area treatment.

Reveal the corner. Negative three times negative three is positive nine. Reversing both strip signs leaves the corner area positive. Five is still present, so the amount needed is still four.

Complete the square. This time its side expression is x minus three. The equation is x minus three, all squared, equals four. The completed area is unchanged, but the side expression remembers the reversed strips.

Take the positive root, two. Removing a signed strip of negative three means adding three. Two plus three gives five. Watch the final subtraction and read the signs together; subtracting a negative quantity increases the result.

Both roots retain the signed shift

Choose the other branch. Negative two plus three is one. The equation with negative six x has solutions five and one, while the earlier equation with positive six x had negative one and negative five.

Here is the earlier equation again, with its negative branch visible. Its answer is negative five. This comparison changes the equation as well as its shift; it is not merely a camera change or a different drawing of the same root.

Return to the negative-strip equation and its positive branch, five. Reversing the linear coefficient reflected this pair of roots through zero. That is a comparison of the displayed values; the World is showing the completion, not a plotted reflection.

Gather the construction into the whole view. The negative strips, positive corner, and removal of the negative shift belong to one chain. Keep their signs attached to their mathematical roles as you scan from the source to the result.

Begin below zero

Keep the negative strips but make the constant negative seven. The whole corner still needs positive nine. Starting from negative seven means that sixteen must be added to reach nine. The required addition is a signed difference, nine minus negative seven.

Complete the square by adding sixteen to both sides. The result is x minus three, all squared, equals sixteen. A negative starting constant has led to a positive completed area; those signs describe different quantities.

The positive branch uses four. Remove negative three, and four plus three gives seven. The equal operations on the two sides preserve the source equation throughout.

The other branch uses negative four. Negative four plus three gives negative one. These two roots, seven and negative one, are equally spaced around positive three. The strip sign sets that central shift.

Signed strips with no constant corner

Try x squared minus two x equals zero. There is no constant term. Split negative two x into two strips of negative one x each. Their labels carry the signs; the drawing still uses ordinary positive screen distances to represent those algebraic contributions.

The two signed widths multiply to positive one, so the full corner is one. With no constant already present, one must be added to each side. A negative strip does not make its square negative. Here negative one times negative one supplies the positive corner.

Complete the square to obtain x minus one, all squared, equals one. Notice the different signs: the side contains minus one, while the corner and completed right-hand area are positive one. These quantities have different mathematical roles and should not be assigned one common sign.

The positive branch gives one. Removing the strip width of negative one means adding one, so the solution is two. Read the operation as one minus negative one. Keeping the signed width intact explains the addition instead of requiring a separate rule for this example.

The negative branch gives negative one. Add one and the solution is zero. Thus an equation with a missing constant corner can have zero as one of its roots. Nothing was lost when the corner began empty; its absence was a meaningful part of the source equation.

Change the equation or change the branch

Reverse only the linear coefficient to positive two x, keeping the constant zero. The strips now have width positive one, and their corner is still positive one. Completion reads x plus one squared equals one. The changed strip sign changes the side expression, even though the completed area remains the same.

Its positive branch gives one minus one, which is zero. The earlier negative-strip equation also had a zero root, but that root came from its negative branch. Branch labels are tied to the completed side expression, so a shared final answer need not occupy the same branch in different equations.

The negative branch gives negative one minus one, or negative two. The two equations therefore have root pairs zero and two, and zero and negative two. Reversing the linear term reversed the nonzero root. We are comparing displayed algebraic values; the World is not drawing a reflection diagram.

Return to x squared minus two x and its positive branch, two. Changing a branch leaves the original coefficients fixed; reversing a coefficient changes the equation itself. Keeping those operations separate prevents the plus-or-minus choice from being mistaken for permission to change a sign anywhere in the equation.

A larger negative constant

Consider x squared minus eight x minus nine equals zero. Each strip has signed width negative four. Their full corner is positive sixteen. Because the existing constant is negative nine, the gap to positive sixteen is twenty-five, not seven. We must account for the negative starting contribution as well as the positive corner.

Add twenty-five to both sides. On the left, negative nine plus twenty-five supplies the needed sixteen. The completed equation is x minus four squared equals twenty-five. The same addition appears on the right because the source right-hand side was zero.

The positive square root is five. Remove the signed width negative four, which means add four, giving x equals nine. There are two different sign operations here: squaring the strip width made a positive corner; subtracting that negative width now produces a positive shift in the solution.

The negative square root is negative five. Adding four gives negative one. Both solutions belong to the same equation. A negative linear term does not force both roots to be positive, just as a positive square-root branch does not always force its final answer to be positive.

The whole view connects the negative strips, negative starting constant, positive corner and both-root procedure. Signs are attached to particular quantities and operations. Following those roles is the reliable route through a signed completion; the colors and labels support the algebra without inventing negative physical lengths.

A smaller signed example

Try x squared minus four x plus three equals zero. Predict the strip width before the corner appears. Sharing negative four x makes two strips of negative two x.

The corner is positive four. Three is already present, leaving one. Complete the square: x minus two, all squared, equals one. Neither a negative strip nor a positive constant changes this same corner comparison.

The positive branch gives one plus two, or three. To predict the other branch, keep the same removal of negative two and change only the sign of the root of one.

The negative branch gives negative one plus two, or one. Signs now have specific jobs: the strips determine the shift, their product determines the corner, and the branch determines which signed root is used.