Square Roots and Completing the Square

6.2: Square Roots and Completing the Square. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 83 parts Practice Flashcards

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Square-root equations

Objective: Solve an equation with a squared binomial.

Prerequisite skills: square roots, factoring, and linear equations. If (x − h)² = k with k > 0, take both positive and negative square roots. If k = 0 there is one distinct real solution; if k < 0 there is none over the reals. Common error: taking only the positive square-root branch. Restrictions and interpretation: Include both square-root signs when solving a square equal to a positive number; keep real-domain conditions.

Formulas

$(x − h)² = k ⇒ x = h ± √k for k ≥ 0$

Worked examples

Try itWhat is the greater real solution of (x − 23)² = 576?Show answer

Take both branches: x − 23 = ±24. Thus x = -1 or 47; the greater is 47. Therefore the requested answer is 47.

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Square-root equations

Objective: Solve an equation with a squared binomial.

Formulas

$(x − h)² = k ⇒ x = h ± √k for k ≥ 0$

Worked examples

Try itWhat is the greater real solution of (x − 23)² = 576?Show answer

Take both branches: x − 23 = ±24. Thus x = -1 or 47; the greater is 47. Therefore the requested answer is 47.

Complete a monic square

Objective: Choose the constant that creates a perfect square.

For x² + bx, add (b/2)² to form (x + b/2)². If adding it within an equation, add it to both sides. Completing the square exposes the vertex and provides a solving method when factoring is inconvenient. Common error: squaring b rather than b/2. Restrictions and interpretation: Include both square-root signs when solving a square equal to a positive number; keep real-domain conditions.

Formulas

$x² + bx + (b/2)² = (x + b/2)²$

Worked examples

Try itWhat k makes x² + 48x + k a perfect square of the form (x + a)²?Show answer

Half the x-coefficient 48 is 24. Square it: k = 24² = 576. The expression becomes (x + 24)². Therefore the requested answer is 576.

Complete the square to solve

Objective: Shift the constant and isolate a square in a nonfactored equation.

Move the constant to the other side, add the square of half the linear coefficient to both sides, and rewrite the left side as a square. Take both root branches and translate back. Common error: adding the completing constant on only one side. Restrictions and interpretation: Include both square-root signs when solving a square equal to a positive number; keep real-domain conditions. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$x² + 2hx = c ⇒ (x + h)² = c + h²$

Worked examples

Try itWhat is the greater solution of x² + 50x = 104?Show answer

Add 625 to both sides: (x + 25)² = 729 = 729. Thus x + 25 = ±27. The greater solution is 27 − 25 = 2. Therefore the requested answer is 2.

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