Question 1

What are the steps for completing the square when a = 1?

Accepted Answer

For x² + bx + c: Step 1 — move c to the right: x² + bx = −c. Step 2 — add (b/2)² to both sides: x² + bx + (b/2)² = (b/2)² − c. Step 3 — write the left side as a perfect square: (x + b/2)² = (b/2)² − c. Step 4 — take square roots and solve for x. The vertex form is (x + b/2)² + (c − (b/2)²), so h = −b/2, k = c − b²/4.

Question 2

How do you complete the square when a ≠ 1?

Accepted Answer

When a ≠ 1, first divide (or factor out) a from the x terms only. Example: 2x² + 12x + 5. Factor 2: 2(x² + 6x) + 5. Complete the square inside: half of 6 is 3, 3² = 9. Add and subtract 9 inside: 2(x² + 6x + 9 − 9) + 5 = 2(x+3)² − 18 + 5 = 2(x+3)² − 13. Vertex: (−3, −13). This step — balancing the added term by the factor a — is the most common mistake.

Question 3

How does completing the square derive the quadratic formula?

Accepted Answer

Start with ax² + bx + c = 0. Divide by a: x² + (b/a)x + c/a = 0. Move c/a: x² + (b/a)x = −c/a. Add (b/2a)²: x² + (b/a)x + (b/2a)² = (b/2a)² − c/a. Left side: (x + b/2a)². Right side: (b² − 4ac)/4a². Take square root: x + b/2a = ±√(b²−4ac)/2a. Solve: x = (−b ± √(b²−4ac))/2a. This is the quadratic formula — completing the square is its proof.

Question 4

When is completing the square more useful than the quadratic formula?

Accepted Answer

Completing the square is preferred when: (1) you need the vertex form for graphing the parabola; (2) a = 1 and b is even, making the arithmetic simple; (3) you are studying calculus and need to integrate expressions like 1/(x²+2x+5) — you complete the square first to use a standard integral. The quadratic formula is better when you just need the roots quickly, especially for non-integer coefficients.

Question 5

How do I find the roots from vertex form?

Accepted Answer

From a(x−h)² + k = 0: rearrange to (x−h)² = −k/a. Take square roots: x − h = ±√(−k/a). Solve: x = h ± √(−k/a). Roots exist (are real) only if −k/a ≥ 0. If a > 0, roots are real when k ≤ 0. If a < 0, roots are real when k ≥ 0. This connects the vertex position directly to whether the parabola crosses the x-axis.

Question 6

What is the axis of symmetry and how is it found?

Accepted Answer

The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is x = h = −b/2a. Any two points on the parabola equidistant from the axis have the same y-value. The axis of symmetry is also the average of the two x-intercepts when they exist: x = (r₁ + r₂)/2 = −b/2a by Vieta's formulas.

Standard Form	Vertex Form	Vertex (h,k)	Roots
x² + 6x + 5	(x+3)² − 4	(−3, −4)	x = −1, −5
x² − 4x + 4	(x−2)²	(2, 0)	x = 2 (repeated)
x² + 2x + 5	(x+1)² + 4	(−1, 4)	No real roots
2x² + 12x + 5	2(x+3)² − 13	(−3, −13)	x ≈ −0.46, −5.54
x² − 6x + 9	(x−3)²	(3, 0)	x = 3 (repeated)

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