Step by Step Solver: Complete the Square

Completing the square rewrites a quadratic expression \( ax^2 + bx + c \) in vertex form \( a(x-h)^2 + k \).

We use the identities:

\( (x + \frac{b}{2})^2 = x^2 + bx + (\frac{b}{2})^2 \)   ⟹   \( x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2 \)

The vertex of the parabola is at \( (h, k) \). Exact fractional values and simplified radicals are used throughout.

✧ Complete the Square ✧

Enter coefficients a, b, c of the quadratic y = ax² + bx + c

Quadratic Expression: \( y = ax^2 + bx + c \)

\( y = \) \( 2x^2 - 4x + 1 \)

⚠️ If a = 0, it will be set to 1. Exact fractions and simplified radicals.
📐 Vertex Form: \( y = a(x-h)^2 + k \)
\( y = 2(x-1)^2 - 1 \)
📍 Vertex: \( (h, k) = \) \( (1, -1) \)

📖 Step-by-Step Solution

💡 Identity used: \( x^2 + bx = (x + \frac{b}{2})^2 - (\frac{b}{2})^2 \)
STEP 1: Isolate the x² and x terms on one side
\( y - 1 = 2x^2 - 4x \)
STEP 2: Divide both sides by the coefficient of x²
\( \frac{y - 1}{2} = x^2 - 2x \)
STEP 3: Complete the square on the right side using the identity
\( x^2 - 2x = (x - 1)^2 - 1 \)
STEP 4: Substitute back and rewrite the equation
\( \frac{y - 1}{2} = (x - 1)^2 - 1 \)
STEP 5: Multiply and simplify to vertex form \( y = a(x-h)^2 + k \)
\( y = 2(x - 1)^2 - 1 \)
💡 Interpretation
The vertex of the parabola is at \( (h, k) \). The axis of symmetry is \( x = h \). The parabola opens upward if \( a > 0 \), downward if \( a < 0 \).
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