Completing squares in two variables

Textbook reading

The problem that opens the lesson

Classify and graph $x^2+6x+4y^2-8y=11$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed. The relevant conditions are not optional bookkeeping: If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully. Following that structure gives $(x+3)^2+4(y-1)^2=24,$ an ellipse.

Why this works

Both variable groups and the equation’s right side must remain balanced. The final constant determines scale and may reveal a degenerate or empty relation. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

Completing the square rewrites quadratic expressions so translated centers, vertices, and axes become visible.

For $x^2+Bx,$ add $(\frac{B}{2})^2$ inside a balanced equation. When a coefficient multiplies $x^2$ or $y^2,$ factor it from the entire variable group before completing the square.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

Both variable groups and the equation’s right side must remain balanced. The final constant determines scale and may reveal a degenerate or empty relation.

Textbook reading

A reliable way to work

Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed.

If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is adding a completion term inside a factored group without multiplying its contribution correctly on the other side.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

Classify and graph $x^2+6x+4y^2-8y=11$ .

Solution

Begin by identifying the mathematical object and the information that fixes it. Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed. The relevant conditions are not optional bookkeeping: If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully. Following that structure gives $(x+3)^2+4(y-1)^2=24,$ an ellipse.

Why this works

Both variable groups and the equation’s right side must remain balanced. The final constant determines scale and may reveal a degenerate or empty relation. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Complete squares while balancing both sides.

Worked development

Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. For $x^2+Bx,$ add $(\frac{B}{2})^2$ inside a balanced equation. When a coefficient multiplies $x^2$ or $y^2,$ factor it from the entire variable group before completing the square. Then apply the conditions explicitly: If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The technique connects general quadratic equations to geometric conic parameters.

Reasoning example

Problem

Factor coefficients before completing.

Worked development

Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. For $x^2+Bx,$ add $(\frac{B}{2})^2$ inside a balanced equation. When a coefficient multiplies $x^2$ or $y^2,$ factor it from the entire variable group before completing the square. Then apply the conditions explicitly: If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The technique connects general quadratic equations to geometric conic parameters.

Worked example 4: quick check

Put $x^2-4x-y^2-6y=12$ in standard form.

Solution

Begin by identifying the mathematical object and the information that fixes it. Collect $x$ and $y$ terms, move constants, factor leading coefficients, complete each square, combine constants, and normalize the right side when needed. The relevant conditions are not optional bookkeeping: If a squared coefficient is negative, factoring changes the sign of the completion constant and must be tracked carefully. Following that structure gives $(x-2)^2-(y+3)^2=7$ .

Why this works

Both variable groups and the equation’s right side must remain balanced. The final constant determines scale and may reveal a degenerate or empty relation. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

Application and interpretation

The technique connects general quadratic equations to geometric conic parameters.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Practice