What is completing the square?

Completing the square is a geometric and algebraic technique that converts a quadratic expression ax² + bx + c into a perfect square form a(x - h)² + k.

What does adding (b/2)² do visually?

It fills the missing corner square of a geometric layout made of an x*x square and two (b/2)*x rectangles, creating a complete large square.

Completing the Square Visualizer - Quadratic Solutions

Splitting b/2 term3

Completed Corner Area (b/2)²9

Calculated Roots (x) x ≈ 2.0, x ≈ -8.0

Variable Adjuster

Visual Variable Value (x)5

Coefficient b (bx) - Even values preferred6

210

Constant c (Target Area)16

140

Completing the Square

SQCOM

Completing the square is a geometric technique to solve quadratic equations. We model x^2 as a square of side x, and bx as two side rectangles of width b/2. Adding a small corner square of area (b/2)^2 completes the shape into a larger perfect square.

x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2

Whiteboard Solver Steps

Step 1

Original Equation

We begin with our quadratic equation in the form $x^2 + bx = c$ , where we have an unknown square of side length $x$ (area $x^2$ ) plus $b$ rectangular strips of width $1$ and height $x$ (total area 6x $), equating to a target area of 16.$

x^2 + 6x = 16

Step 2

Split the Linear Coefficient (Geometric Alignment)

To maintain symmetry and build towards a perfect square shape, we divide the linear coefficient ( $b = 6$ ) in half, giving $b/2 = 3$ . Geometrically, we split the area $bx$ into two equal rectangles of dimensions $x \times 3$ and place one on the right side of the main $x^2$ square, and the other at the bottom.

\frac{b}{2} = \frac{6}{2} = 3

Step 3

Calculate the Completing Area (Corner Square)

Observe the empty corner that remains at the bottom-right. The dimensions of this missing space are exactly $b/2 \times b/2$ (or 3 \times 3 $). To fill this corner and form a single, large perfect square of side$ (x + b/2) $, we calculate its area:$ (3)^2 = 9 $.$

\left(\frac{b}{2}\right)^2 = 3^2 = 9

Step 4

Add Completing Term to Both Sides (Algebretic Balance)

To preserve algebraic balance, we add the corner area ( $9$ ) to both sides of the equation. The left side is now a perfect trinomial square: $x^2 + 6x + 9 = (x + 3)^2$ . The right side becomes $c + (b/2)^2 = 16 + 9 = 25$ .

\begin{aligned}x^2 + 6x + 9 = 16 + 9 \\ (x + 3)^2 = 25\end{aligned}

Step 5

Take the Square Root

To isolate $x$ , we take the square root of both sides. This yields two possibilities: $x + 3 = \sqrt{25}$ and $x + 3 = -\sqrt{25}$ .

x + 3 = \pm \sqrt{25} \approx \pm 5.00

Step 6

Solve for x (Find Roots)

Subtract $b/2 = 3$ from both sides to find the solutions. The two real roots represent the values of $x$ where the equation balances, giving $x_1 \approx 2$ and $x_2 \approx -8$ .

\begin{aligned}x_1 = -3 + 5.00 = 2 \\ x_2 = -3 - 5.00 = -8\end{aligned}

RV Anveshana